aa r X i v : . [ m a t h . HO ] D ec ARBITRARILY CLOSE
JOHN A. ROCK
This paper is dedicated to y and B.
Abstract.
Mathematicians tend to use the phrase “arbitrarily close” to meansomething along the lines of “every neighborhood of a point intersects a set”.Taking the latter phrase as a technical definition for arbitrarily close leads toan alternative, or at least parallel, development of classical concepts in analysissuch as closure and limits in the context of metric spaces as well as continuity,differentiation, and integration in the setting of real valued functions on thereal line. In particular, a definition of integration in terms of arbitrarily closeis present here. The corresponding integral is distinct from and yet equivalentto the classical integrals of Riemann and Darboux. INTRODUCTION AND OVERVIEW
In a first calculus course, many students are confronted with the twin notions oflimit and convergence but often with technical definitions subdued or fully ignored.Whether in terms of sequences or functions, the definition of limit is, frankly, dif-ficult to understand. Once these students reach their first undergraduate course inanalysis, it is not easy to develop a technical definition of limit following an intuitivediscussion along the lines of what it means for a sequence or function to “approach”a given value, a heuristic phrase they may have heard in a calculus course. Thus,one of the primary motivations of this paper is to provide a formal definition of ar-bitrarily close (Definition 2.2) and, in turn, the accretion of a sequence or a function(Definitions 6.4 and 6.4, respectively) in order to—with any luck—serve as a moreintuitive but technically sound framework for undergraduate analysis. Ultimately,the goal is to provide a development of many of the fundamental results in analysisbased on a concrete notion of what it means for a point to be arbitrarily close to aset.As something of an overview, several of the ideas explored in this paper includeprecise versions of the following statements: • Zero is arbitrarily close to the set of positive and the set of negative realnumbers. See Lemma 2.6. • A sequential limit is arbitrarily close to its sequence. See Theorem 2.9. • A closed set contains the points arbitrarily close to the set. See Definition4.1 and Theorem 4.4.
Date : January 1, 2020.2010
Mathematics Subject Classification.
Primary 26A03, 26A06, 26A42, 97E40; Secondary11B05, 54A20, 97I30, 97I40, 97I50.
Key words and phrases. arbitrarily close, accretion, closed, open, supremum, infimum, limit,continuity, differentiation, integration. • A functional limit is arbitrarily close to the range of the function. SeeTheorem 6.12. • A derivative is arbitrarily close to the set of difference quotients. See Corol-lary 7.7. • An integral is arbitrarily close to the sets of upper and lower sums (whichare more general than Darboux sums here). See Definition 8.3 and Theorem8.8.The next section develops the main topic of the paper: A formal definition ofthe phrase arbitrarily close .2.
THE DEFINITION OF ARBITRARILY CLOSE
What is meant—mathematically—when two objects are said to be arbitrarilyclose ? To set the stage for a technical definition of this concept, consider thedefinition of a metric space which allows for a specific meaning for the distancebetween pairs of points.
Definition 2.1. A metric space is a nonempty set X paired with a function d ,denoted by ( X, d ), where d : X × X → R such that:(i) d ( x, y ) ≥ x and y in X ;(ii) d ( x, y ) = 0 if and only if x = y ;(iii) d ( x, y ) = d ( y, x ) for every pair of points x and y in X ; and(iv) for any three points x, y, and z in X it follows that d ( x, z ) + d ( z, y ) ≤ d ( x, y ) . (2.1)Inequality (2.1) is known as the triangle inequality . Given a pair of points x and y in X , the nonnegative real number d ( x, y ) is called the distance between x and y .The notion of distance provided by a metric space allows for the considerationof the more subtle notion of how close a point is to a set. In particular, perhapsthe following definition does justice to intuition regarding what it might mean fora point to be arbitrarily close to a set. Definition 2.2.
Let (
X, d ) be a metric space with y ∈ X and B ⊆ X . The point y is said to be arbitrarily close to B , or B is arbitrarily close to y , if for every ε > x ∈ B such that d ( x, y ) < ε . In this case, y acl B is written. In thecase where some z ∈ X is not arbitrarily close to B , z nacl B is written.In a metric space ( X, d ), the ε - neighborhood centered a point c of a given radius(or “error”) ε > V ε ( c ) and defined by V ε ( c ) = { x ∈ X : d ( x, c ) < ε } . (2.2)Thus, y acl B if and only if for every ε > V ε ( c ) ∩ B = ∅ , where ∅ denotes the empty set.In the context of the real line equipped with its standard metric d R , y ∈ R isarbitrarily close B ⊆ R if and only if for every ε > x ∈ B suchthat d R ( x, y ) = | x − y | < ε, or equivalently, y − ε < x < y + ε. (2.3)The right side of Equation (2.3) just above proves to be useful in the developmentof the calculus on the real line throughout the paper. Thanks to Berit Givens for the suggesting this notation.
RBITRARILY CLOSE 3
Remark . A definition of arbitrarily close also holds for topological spaces: In atopological space X , a point y ∈ X is said to be arbitrarily close to a set B ⊆ X ifevery neighborhood of y (that is, every open set containing y ) contains an elementof B . By classical results in topology, this interpretation is equivalent to Definition2.2 in the setting of a metric topology.Definition 2.2 immediately yields the following fact: Points in a set are arbitrarilyclose to the set. Lemma 2.4.
Let ( X, d ) be a metric space with y ∈ X and B ⊆ X . If y ∈ B , then y is arbitrarily close to B .Proof. Assume y ∈ B and let ε >
0. Choosing x = y yields d ( x, y ) = 0 < ε . (cid:3) Lemma 2.4 immediately reveals the fact that in a metric space, a point whichis arbitrarily close to a set is not necessarily an accumulation point of the set. SeeDefinition 6.1 below. Still, these concepts are deeply related and the foundation oftheir relationship is most brightly illuminated by Theorem 2.9 below.Another basic result stemming from Definition 2.2 is the following lemma.
Lemma 2.5.
Let ( X, d ) be a metric space and let x, y ∈ X . Then x = y if andonly if x is arbitrarily close to { y } ( or vice versa ) .Proof. Let x = y . For every ε > d ( x, y ) = 0 < ε . Thus x acl { y } .Now, suppose x = y . Then 0 < ε = d ( x, y ) / ≤ d ( x, y ). Thus x nacl { y } . (cid:3) A intuitive and fairly useful fact is that the only real number arbitrarily close toboth positive and negative real numbers is zero. To this end, let R + = (0 , ∞ ) and R − = ( −∞ , Lemma 2.6.
In the metric space R equipped with its standard metric d R , ℓ = 0 ifand only if ℓ acl R + and ℓ acl R − .Proof. First, assume ℓ = 0 and let ε >
0. Then ε/ ∈ R + , − ε/ ∈ R − , and − ε < − ε/ < < ε/ < ε. (2.4)Therefore ℓ acl R + and ℓ acl R − .Next, without loss of generality, assume ℓ > y <
0. Then y < < ℓ/ < ℓ. (2.5)Since ε = ℓ/ >
0, it follows that d R ( ℓ, y ) = | ℓ − y | > ℓ/
2, and therefore ℓ isnot arbitrarily close to R − . As similar argument shows no negative number isarbitrarily close to R + . (cid:3) In the next example and throughout the paper, the set of positive integers isdenoted by N . Example 2.7.
Consider the open interval I = (0 , ⊆ R . Even though thereal number ℓ = 3140 is not an element of I , ℓ acl I since every ε -neighborhood of3140 contains an element of (0 , ε > x ∈ I can be found where x is less than 3140 but greater than 3140 − ε . For instance,choose x = max { − ε/ , } . Then it follows that both that x is in I and d R ( x, ℓ ) = | x − ℓ | = | x − | ≤ ε/ < ε. (2.6) J. A. ROCK
Since a suitable x in I can be found for each error ε >
0, a sequence in I canbe found that provides a countably infinite collection of elements as close to ℓ asdesired. In particular, x n = 3140 − (1 /n ) is in I for each n ∈ N and for a given ε > n can be taken large enough to get d R ( x n , ℓ ) = | x n − ℓ | = | − (1 /n ) − | = 1 /n < ε. (2.7)The argument just above indicates how deeply the notions of arbitrarily close and limit are related to one another. The definition of limit is arguably the most difficultfor a student of undergraduate mathematics to understand. See [1, Definition 2.2.3,p.43] but also [3, 4, 6] for some other interesting and effective approaches to teachingreal analysis at the undergraduate level which are, in part, designed to address thisdifficulty. Also see [5] for a thorough discussion of the challenges that come withteaching convergence and limits. Hopefully, by first working through the definitionof and results stemming from arbitrarily close, students will be better prepared thefor challenge of understanding convergence.To establish the deep connection between arbitrarily close and limit of a sequence,a definition for the latter is required. Definition 2.8.
Let (
X, d ) be a metric space and for each positive integer n let x n ∈ X . The sequence ( x n ) converges to a point ℓ , also written lim n →∞ x n = ℓ , iffor every ε > N such that for any n ≥ N , d ( x n , ℓ ) < ε. (2.8)In this case ℓ is called the limit of the sequence ( x n ).In other words, lim n →∞ x n = ℓ means that the sequence ( x n ) is arbitrarily closeto its limit ℓ and the sequence can be cut off at some index N so that eventually(that is, for all n ≥ N ) all of the terms x n that appear in the sequence after thecutoff N are as close to ℓ as desired.A considerable difference between the definition of arbitrarily close and that ofconvergence (or limit) is the number of quantifiers. This distinction is perhaps themost important reason to introduce a formal definition of arbitrarily close beforedealing with convergence. Doing so allows for a breakdown and introduction ofconvergence in a couple of steps. With arbitrarily close, one may first focus on themeaning of the quantified statement “for every ε > . . . ” along with the notion ofcloseness given by “ d ( x n , ℓ ) < ε ”. From there, one may tackle the subtle multiply-quantified phrase “there is some N ∈ N such that for any n ≥ N . . . ” appearingin the middle of definition of convergence. The interplay between these quantifiedstatements seems to be a substantial logical barrier for students to overcome. It isexplored further in Section 9.The following theorem precisely establishes the deep connection between thedefinitions of convergence and arbitrarily close. Basically, points arbitrarily closeto a set are limits of sequences of points from the set.
Theorem 2.9.
Let ( X, d ) be a metric space with ℓ ∈ X and S ⊆ X . Then ℓ acl S if and only if there is a sequence ( x n ) of points in S whose limit is ℓ . The proof of Theorem 2.9 is straightforward and relies on nothing more than therelevant definitions. It also makes for a nice exercise at the point where studentsfirst have to deal with the definition of convergence.
RBITRARILY CLOSE 5
Proof.
Assume ℓ is arbitrarily close to S and note that 1 /n > n . So by the definition of arbitrarily close (Definition 2.2), for each n thereis some x n ∈ S where d ( x n , ℓ ) < /n. (2.9)Now, let ε >
0. By the intuitive Archimedean Property (see [1, Theorem 1.4.2,p.21]), there is a positive integer N large enough so that 1 /N < ε . Thus, for every n ≥ N d ( x n , ℓ ) < /n ≤ /N < ε. (2.10)Therefore, ( x n ) ⊆ S and lim n →∞ x n = ℓ .For the converse, assume there is a sequence ( x n ) of points in S whose limit is ℓ and let ε >
0. By the definition of convergence (Definition 2.8), there is somepositive integer N such that x N is in S and d ( x N , ℓ ) < ε . Therefore, ℓ acl S . (cid:3) Be careful. Theorem 2.9 does not say that if a sequence (or rather, the rangeof a sequence) is arbitrarily close to a given real number, then the limit exists andis equal to the given number. In Example 2.10, the range of the sequence ( b n ) isarbitrarily close to two distinct real numbers ℓ = 3141 = 3139 = ℓ and, as itturns out, neither can be the limit due to the uniqueness of limits. Ultimately, thelimit of ( b n ) does not exist. Example 2.10.
Suppose b n = 1 /n + 3140 + ( − n for each n ∈ N . The range S = { b n : n ∈ N } ⊆ R of the sequence ( b n ) is arbitrarily close to both ℓ = 3141and ℓ = 3139. Note that neither ℓ = 3141 nor ℓ = 3139 is in S , yet for any ε > k large enough to so that both | b k − ℓ | = (cid:12)(cid:12)(cid:12)(cid:12) k + 3140 + ( − k − (cid:12)(cid:12)(cid:12)(cid:12) = 12 k < ε (2.11)and | b k +1 − ℓ | = (cid:12)(cid:12)(cid:12)(cid:12) k + 1 + 3140 + ( − k +1 − (cid:12)(cid:12)(cid:12)(cid:12) = 12 k + 1 < ε. (2.12)Thus, the set S is arbitrarily close to both ℓ = 3141 and ℓ = 3139. Be careful.Neither 3141 nor 3139 is the limit of ( b n ) (and neither is in the set S ). It can beproven that the limit of a sequence is unique, so neither lim n →∞ b n = 3141 norlim n →∞ b n = 3139. See the classical result [1, Theorem 2.2.7, p.46], where theproof is left as an exercise and makes use of Lemma 2.5.Before moving on to more connections between the formal definition of arbitrarilyclose and classical results in analysis, consider the following example of a spiral inthe complex plane which is arbitrarily close to, but does not contain points on, theunit circle. Example 2.11.
Let f : [0 , ∞ ) → C be given by f ( t ) = te it / ( t + 1) and for each w ∈ C let | w | denote the modulus of w . Since | f ( t ) | = tt + 1 < t ∈ [0 , ∞ ), the range of f is a spiral in contained V (0), the open unit diskcentered at the origin. Let S denote the spiral given by the range of f . It turns outthat every complex number on the unit circle centered at the origin is arbitrarilyclose to S even though the spiral S and this circle have no points in common. Let J. A. ROCK z = e iθ where θ ∈ [0 , π ) and for each n ∈ N let z n = f ( θ + 2 πn ). Then z is thepoint on the unit circle with argument θ and each z n is a point in the intersectionof the spiral S and the ray connecting the origin to z . Now, let ε > d C denote the standard metric on C . Then for large enough n , d C ( z n , z ) = | z n − z | < ε. (2.14)Thus z acl S .With such a deep connection between arbitrarily close and the vital notion ofconvergence of sequences now established, it should come as no surprise that manyof the concepts explored in analysis can be discussed in terms of points arbitrarilyclose to sets. The next section explores a way in which the definition of arbitrar-ily close provided in Definition 2.2 allows for a more intuitive derivation of thesupremum and infimum of a bounded set of real numbers.3. AN ALTERNATIVE DEFINITION OF SUPREMUM
Consider the following definition for the maximum of a set of real numbers.
Definition 3.1.
A real number b is an upper bound of a set S ⊆ R if x ≤ b forevery x ∈ S . A real number M is the maximum of S if(i) M is an upper bound for S , and(ii) M ∈ S .In this case, M = max S is written. A similar definition holds for minimum .The closed interval F = [0 , F = 3140, butwhat about the open interval I = (0 , I were to have a maximum, it would certainly be 3140. The problem is that while3140 is an upper bound for I , it is not an element of I and no other real numbersatisfies both properties (i) and (ii) from the definition of maximum. Thus, themaximum of I does not exist.Still, the real number ℓ = 3140 plays a special role with regard to the set I =(0 , supremum shows it be like amaximum except it is not necessarily an element of the given set. Simply put, thesupremum of a set is the upper bound arbitrarily close to the set. Definition 3.2.
A real number u is the supremum of a set S ⊆ R if(i) u is an upper bound for S , and(ii) u acl S .In this case, u = sup S is written.An analogous statement holds for the definition of infimum which is denoted byinf S and defined in terms of lower bounds of S . Basically, the infimum of S is thelower bound arbitrarily close to S .Note that in Definition 3.1, part (ii) requires an upper bound M to be in theset in order to have M = max S . In part (ii) of Definition 3.2, the correspondingproperty is relaxed slightly in that an upper bound u need only be arbitrarily closeto the set in order to have u = sup S .As seen in Example 2.7, the real number 3140 is arbitrarily close to open interval I = (0 , I = (0 , { x ∈ R : 0 < x < } , (3.1) RBITRARILY CLOSE 7 so 3140 is also an upper bound for I . Therefore, sup I = 3140.To see that the definition of supremum provided by Definition 3.2 is equivalentto a more classical definition employed by Abbott (see [1, Definition 1.3.2, p.15])—that is, the supremum of a set is its least upper bound —look to a result providedby Abbott (see [1, Lemma 1.3.8, p.17]). The corresponding lemma is stated herefor convenience. Essentially, it says an upper bound is the supremum of a set whenit is arbitrarily close to the set. Lemma 3.3.
Assume u ∈ R is an upper bound for a set A ⊆ R . Then u = sup A if and only if for every choice of ε > , there exists some element a ∈ A satisfying u − ε < a . Assuming u is an upper bound for A , ε >
0, and there exists some element a ∈ A satisfying u − ε < a , it follows that u − ε < a ≤ u < u + ε, (3.2)and therefore d R ( a, u ) = | a − u | < ε . Hence, u = sup A and u is an upper boundarbitrarily close to A . As a result, Definition 3.2 serves as a viable alternative forthe definition of supremum.Analogous results hold for infimum and the proofs follow readily, as with thefollowing corollary. Corollary 3.4.
Suppose S ⊆ R with u = sup S . Then there is a sequence ( x n ) contained in S whose limit is u .Proof. The statement follows readily from Theorem 2.9 and Lemma 3.3. (cid:3)
The technical definition of arbitrarily close in Definition 2.2 not only providesan alternative approach to defining integration and understanding supremum andinfimum, it also immediately connects to the fundamental aspects of metric spaces,especially closed sets.4.
A CLOSED-MINDED APPROACH TO ANALYSIS
The setting and development of closed sets (and open sets) described here isrestricted to metric spaces, but many of the results hold in the more general contextof topological spaces. More importantly, establishing a solid notion of closed setsthrough the definition of arbitrarily close in Definition 2.2 allows for an alternativefoundation for many results in calculus.Throughout this section, let X denote a metric space with metric d . Definition 4.1.
Let B ⊆ X . The closure of B , denoted by B , is the set of pointsarbitrarily close to B . Thus, B = { x ∈ X : x acl B } = { x ∈ X : ∀ ε > , V ε ( x ) ∩ B = ∅ } . (4.1)A set F ⊆ X is closed if F contains the points in X arbitrarily close to F .Definition 4.1 immediately yields a fundamental fact about closed sets. Lemma 4.2.
The intersection of a collection of closed sets is closed.Proof.
Let { F α } denote a collection of closed sets in some metric space and suppose y is arbitrarily close to ∩ α F α . Let ε > ε -neighborhood of V ε ( y ).Then there is some x ∈ ( ∩ α F α ) ∩ V ε ( y ) which means x ∈ F α ∩ V ε ( y ) for each α . J. A. ROCK
Since ε was chosen arbitrarily and each F α is closed, it follows that y ∈ F α for each α , thus y ∈ ∩ α F α . Therefore, ∩ α F α is closed. (cid:3) Any point z which is not in a closed set F is by Definition 4.1 is not arbitrarilyclose to F . Hence, there is an ε -neighborhood V ε ( z ) of z which does not intersect F . This is precisely a characterizing property—therefore a defining property—ofpoints in an open set (cf. [1, Definition 3.2.1, p.88]). To this end, the complementof a set S in a metric space X is denoted by S c and given by S c = X \ S = { x ∈ X : x / ∈ S } . (4.2)Also, a point y is not arbitrarily close to a set S if there is some ε > x ∈ S satisfies d ( x, y ) ≥ ε . (4.3)In other words, for some positive distance ε , every x ∈ S is at least ε away from y . Equivalently, y nacl S if and only if there is some ε -neighborhood of y that doesnot intersect S . That is, V ε ( y ) ∩ S = ∅ for some ε > Definition 4.3.
A set O ⊆ X is open if for every a ∈ O there is an ε -neighborhoodof a contained in O (that is, V ε ( a ) ⊆ O for some ε > Theorem 4.4.
A set O ⊆ X is open if and only if O c is closed. A proof very similar to the one presented here was created by a student asshe prepared for a final exam in the summer of 2019. In particular, she used thelanguage of arbitrarily close and preferred this approach over the one used in [1,Theorem 3.2.13, p.92].
Proof.
Assume O is open and suppose y is arbitrarily close to O c . By way ofcontradiction, assume y ∈ O . Since O is open, there is an ε -neighborhood of y contained in O . This means every x ∈ O c lives outside of this ε -neighborhood of y , thus x is at least a positive distance ε away from y . Hence, y is not arbitrarilyclose to O c , a contradiction. Therefore, O c is closed.For the converse, assume O c is closed and let z ∈ O . Since O c contains all pointsarbitrarily close to O c , z is not one of them. Since z is not arbitrarily close to O c ,there must be some ε z > V ε z ( z ) ⊆ O . Therefore, O is open. (cid:3) The following is just a concrete example illustrating Theorem 4.4.
Example 4.5.
Consider the closed interval F = [0 , w = 3141. Also,consider the positive distance ε = 1 /
10. Then for every x ∈ F it follows that | x − | ≥ /
10 = ε . (4.4)As a result, the neighborhood V / (3141) = (3141 − / , /
10) contains w and is completely away from F in the sense that[0 , ∩ (3141 − / , /
10) = ∅ . (4.5)Now consider the complement F c = R \ F = ( −∞ , ∪ (3140 , ∞ ). Every elementin R \ F comes with an ε -neighborhood that is also contained in R \ F . Specifically,for each z ∈ R \ F , define ε z to be the shorter of the distances between z and the RBITRARILY CLOSE 9 endpoints of F = [0 , ε z = min {| z − | , | z − |} and ε z > z = 0and z = 3140. Then V ε z ( z ) ∩ [0 , ∅ . (4.6)Therefore, V ε z ( z ) ⊆ R \ F .That is, every element in the complement of the closed interval F comes withan ε -neighborhood that is also contained in the complement F c = R \ F , which isopen by Theorem 4.4.Metric spaces provide a natural setting for a notion of boundedness. Definition 4.6.
Let (
X, d ) denote a metric space. A set A ⊆ X is said to be bounded if there is a point z in X a nonnegative real number u such that d ( z, x ) ≤ u for all x ∈ A. (4.7)Thanks to the triangle inequality (2.1), the choice of the point z as above hasno effect on whether or not a given set is bounded. In the context of the real line R or the complex plane C equipped with their standard metrics, the choice z = 0is made throughout the paper. Lemma 4.7.
Suppose A is bounded in a metric space ( X, d ) with z in X and u ≥ such that d ( z, x ) ≤ u for every point x in A . Then A is bounded and d ( z, y ) ≤ u for every point y in A .Proof. Suppose A is bounded with z in X and u ≥ d ( z, x ) ≤ u for everypoint x in A . Let y ∈ A and ε >
0. Since y acl A , there is some a in A such that d ( a, y ) < ε . Paired with the triangle inequality (2.1), it follows that d ( z, y ) ≤ d ( z, a ) + d ( a, y ) < u + ε. (4.8)Since ε > A is bounded and d ( z, y ) ≤ u for every point y in A . (cid:3) The boundary of a set is another topological concept that readily lends itselfto an alternative—and perhaps more intuitive—definition. For comparison, see [2,Definition 2.13, p.65] but also the discussions on page 9 where the phrase “arbi-trarily close” is used but not defined, and page 65 regarding “points that lie closeto both the inside and the outside of the set”.
Definition 4.8.
The boundary of a set S ⊆ X , denoted by ∂S , is the set of pointsarbitrarily close to both S and its complement S c .Again, this definition of boundary is intuitive. Points on the boundary of a setare near both the set and the complement of the set. Such points live in the closure,but not the interior, of the set. Definition 4.9.
Let S ⊆ X . The interior of S , denoted by ˚ S , is the union of allthe open sets contained in S . That is, ˚ S is the largest open set contained in S .The next theorem follows from the fact that each point in an open set comeswith an ε -neighborhood which is also contained in the set. Theorem 4.10.
For any set S ⊆ X , ∂S = S \ ˚ S . Proof.
Assume y ∈ ∂S . Then y acl S and y acl S c . So, y ∈ S and for every ε > V ε ( y ) ∩ S c = ∅ . Hence, no ε -neighborhood of y is contained in S , so y / ∈ ˚ S .For the converse, assume y ∈ S \ ˚ S . Since y ∈ S it follows that y acl S . But since y / ∈ ˚ S , for every ε > V ε ( y ) ∩ S c = ∅ . Thus y acl S c as well. (cid:3) To conclude this section, consider the example of the topologist’s sine curverealized in the complex plane.
Example 4.11.
Let S ⊆ C be the range of the function g : R + → C given by g ( t ) = t + i sin (cid:18) t (cid:19) (4.9)and let T = { iy : − ≤ y ≤ } , which is the closed line segment connecting − i and i in the complex plane. Even though S ∩ T = ∅ , it follows that T ⊆ ∂S . Indeed,given any point in T there is a convergent sequence of points in the range of g andanother convergent sequence outside of the range of g whose limits are the givenpoint.Let z ∈ T and ε >
0. Then z = iy for some y ∈ [ − , ω ∈ [0 , π ) such that sin ω = y . For each n ∈ N , define t n = 1 / ( ω + 2 πn ). Thus,for any ε > n ≥ N where N > π (cid:18) ε − ω (cid:19) , (4.10)it follows that d C ( g ( t n ) , z ) = | g ( t n ) − z | = t n = 1 ω + 2 πn ≤ ω + 2 πN < ε. (4.11)Therefore, lim n →∞ g ( t n ) = z . By Theorem 2.9, it follows that z acl S .Now consider the sequence ( z − / ( ω + 2 πn )). Each point in this sequence isin S c . For ε and n ≥ N as above it follows that z − / ( ω + 2 πn ) is in V ε ( z ) ∩ S c .Therefore, z acl S c as well. By Theorem 4.10 it follows that z ∈ ∂S .Sequences have already played a prominent role in the development of the paperthus far, but the following section explores properties of sequences through the lensof arbitrarily close by taking closures of tails.5. ACCRETION OF SEQUENCES
What is meant when a sequence is said to “approach” a given point? In calculustexts, a phrase such as “the sequence ( x n ) approaches ℓ ” may be taken to meanlim n →∞ x n = ℓ . But such a correspondence merely ties a vague phrase to a verytechnical statement, leaving a lot to be desired. The setting provided by gatheringall points arbitrarily close to a given sequence provides an intermediate step towardsunderstanding limits of sequences. Definition 5.1.
Let ( x n ) be a sequence of points in a metric space. The accretionof ( x n ), denoted by A (( x n )), is defined by A (( x n )) = \ N ∈ N ∪{ } { x N + n : n ∈ N } . (5.1)That is, the accretion of a sequence is the intersection of the closure of the tailsof the sequence. To foster intuition, it helps to unpack Definition 5.1 as follows: RBITRARILY CLOSE 11 (1) First, for each N ∈ N ∪ { } , the N -tail (or simply tail ) of the sequence ( x n )is the subsequence ( x N + n ) whose first term is x N +1 and which is a copy ofthe sequence ( x n ) thereafter.(2) Taking the closure of each N -tail, denoted by { x N + n : n ∈ N } above, picksup all points arbitrarily close to the N -tail (technically, all points arbitrarilyclose to the range of the corresponding subsequence).(3) Finally, the intersection of the closures of all N -tails, denoted above by \ N ∈ N ∪{ } { x N + n : n ∈ N } , (5.2) amounts to gathering all points arbitrarily close to the sequence ( x n ) andtruncating terms which are not arbitrarily close to each of the N -tails.There is an immediate corollary of Lemma 4.2 since A (( x n )) is an intersectionof closed sets. Corollary 5.2.
The accretion of a sequence is closed.
The structure of the accretion of a sequence can vary considerably, as the fol-lowing examples show.
Example 5.3.
Consider the sequence of real numbers given by ( n ), the enumera-tion of the natural numbers. Then for each N ∈ N ∪ { } it follows that A (( n )) = \ N ∈ N ∪{ } { N + n : n ∈ N } = \ N ∈ N ∪{ } { N + 1 , N + 2 , . . . } = ∅ . (5.3)Thus, it is possible for the accretion of a sequence to be empty. Example 5.4.
Let ( z n ) be the sequence of complex numbers with terms definedfor each n ∈ N by z n = i n ( n − /n = i n − ( i n /n ). Then A (( z n )) = \ N ∈ N ∪{ } { i N + n − ( i N + n / ( N + n )) : n ∈ N } (5.4) = { i k : k = 0 , , , } = { , i, − , − i } . (5.5)Indeed, for each i k where k = 0 , , , ε >
0, there is a natural number m large enough so that | z m + k − i k | = 1 / (4 m + k ) < ε. (5.6)Thus, for each k = 0 , , , N ∈ N ∪ { } it follows that i k acl { i N + n − ( i N + n / ( N + n )) : n ∈ N } . (5.7)No other complex number is arbitrarily close to each N -tail of the sequence ( z n ). Example 5.5.
Let ( q n ) be an enumeration of the rational numbers in the closedunit interval [0 , f : N → Q ∩ [0 ,
1] where f ( n ) = q n for each natural number n . Then by the density of the rational numbers in the realnumbers, for each N ∈ N ∪ { } it follows that { q N + n : n ∈ N } = [0 ,
1] despite thefact that the first N terms have been left out of corresponding tail of the sequence.Therefore, A (( q n )) = [0 , Q ∩ [0 ,
1] = [0 , The next example shows that care must be taken when connecting the notionsof accretion and limit of a sequence. It is not enough to suggest that the accretionneed only be singleton in order for the limit of the sequence to exist.
Example 5.6.
Let ( b n ) be the sequence of real numbers with terms defined by b n = 0 when n is odd and b n = n when n is even. Then for each N ∈ N ∪ { } andthe corresponding N -tail it follows that { b N + n : n ∈ N } ⊆ { } ∪ { N + 1 , N + 2 , . . . } .Hence, { } ⊆ A (( b n )) ⊆ \ N ∈ N ∪{ } ( { } ∪ { N + 1 , N + 2 , . . . } ) = { } . (5.8)Therefore, A (( b n )) = { } , but as an unbounded sequence ( b n ) does not converge.The following proposition provides a feature of convergent sequences in the con-text of accretion. Proposition 5.7.
If a sequence of points ( x n ) in a metric space ( X, d ) converges,then its accretion A (( x n )) is a singleton and A (( x n )) = n lim n →∞ x n o . (5.9) Proof.
Assume lim n →∞ x n = x and let ε >
0. There is some positive integer K such that for every positive integer k ≥ K it follows that d ( x k , x ) < ε. So for every N ∈ N ∪ { } it follows that x ∈ { x N + n : n ∈ N } . (5.10)Hence, lim n →∞ x n = x ∈ A (( x n )).Now suppose y = x . Then d ( x, y ) > N ∈ N such that forall k ≥ N it follows that d ( x, y ) ≤ d ( x, x k ) + d ( x k , y ) < d ( x, y ) + d ( x k , y ) , (5.11)and therefore, 12 d ( x, y ) ≤ d ( x k , y ) . (5.12)Thus, y / ∈ { x N + n : n ∈ N } and so y / ∈ A (( x n )). (cid:3) A definition for subsequences is needed before concluding the section.
Definition 5.8.
Let ( x n ) be a sequence of points in a metric space and let n Let ( x n ) be a sequence of points in a metric space. Then A (( x n )) = L (( x n )) . RBITRARILY CLOSE 13 Proof. This result follows from Theorem 2.9 and the fact that a subsequence isitself a sequence. (cid:3) The accretion of a sequence connects directly to the classical notions of limitsuperior and limit inferior for bounded sequences of real numbers. Definition 5.10. Let ( x n ) be a bounded sequence of real numbers. Then the limitsuperior and limit inferior of ( x n ), denoted by lim sup x n and lim inf x n , respec-tively, are defined bylim sup x n = sup L (( x n )) and lim inf x n = inf L (( x n )) . (5.14)The next corollary follows immediately from a string of results obtained thusfar. Essentially, the limit superior and limit inferior of a bounded sequence ofreal numbers are the maximum and minimum, respectively, of the accretion of thesequence. Corollary 5.11. If ( x n ) is a bounded sequence of real numbers, then lim sup x n ∈ L (( x n )) and lim inf x n ∈ L (( x n )) . (5.15) Equivalently, lim sup x n = max A (( x n )) and lim inf x n = min A (( x n )) . (5.16) Proof. The proof follows from Proposition 5.9, Corollary 5.2, Lemma 4.7 (to ensurethe corresponding supremum and infimum exist), and Corollary 3.4 (and its analogfor infima). (cid:3) The section concludes with the following characterization of convergent sequencesof real numbers. Corollary 5.12. Suppose ( x n ) is a sequence of real numbers. Then lim n →∞ x n exists if and only if the accretion A (( x n )) is a singleton and ( x n ) is bounded.Proof. If lim n →∞ x n exists, then ( x n ) is bounded (see [1, Theorem 2.3.2, p.49]) andby Proposition 5.7 is a singleton.On the other hand, if A (( x n )) is a singleton and ( x n ) is bounded, then byCorollary 5.11 it follows that lim sup x n = lim inf x n and so lim n →∞ x n exists. (cid:3) The notion of accretion for functions defined in the next section allows for aparallel approach to functional limits and pointwise continuity.6. ACCRETION AND LIMITS OF FUNCTIONS The definition of the limit of a function is another difficult concept for newmathematicians to fully understand, as indicated in [3, 4, 5, 6] and as can becorroborated by just about any undergraduate mathematics major. As such, thissection is designed to face the challenge of understanding limits and pointwisecontinuity of functions by providing an alternative development in the context ofaccretion.In order to align with a more classical approach to analysis, consider the followingdefinition. Definition 6.1. Let ( X, d ) denote a metric space. A point c ∈ X is an accumulationpoint of a set A ⊆ X if c acl ( A \ { c } ). That is, c is an accumulation point of A iffor every ε > V ε ( ℓ ) ∩ ( A \ { c } ) = ∅ . In other words, c is an accumulation point of A if every ε -neighborhood of c intersects A at some point other than c .In [1, 2] and other undergraduate textbooks on analysis, accumulation points arealso called “cluster points” or “limit points”. For instance, see [1, Definition 3.2.4,p.89] and [2, Definition 2.7, p.60]. Such terminology is justified since accumulationpoints of a set are limits of sequences in the set. See [1, Theorem 3.2.5, p.89], thestatement of which is provided in the following corollary of Theorem 2.9. Corollary 6.2. Let ( X, d ) denote a metric space. A point x ∈ X is an accumulationpoint of a set A ⊆ X if and only if x = lim n →∞ a n for some sequence ( a n ) of pointsin A with x = a n for every n ∈ N .Proof. The statement is a special case of Theorem 2.9. Points arbitrarily close toa set are limits of sequences of points contained in the set. (cid:3) Here is a classical definition of the limit of a function from one metric space toanother. Definition 6.3. Let ( X, d X ) and ( Y, d Y ) be metric spaces. Let A be a subset of X , let f : A → Y , and let c be an accumulation point of A . A point ℓ ∈ Y is the limit of f at c , written lim x → c f ( x ) = ℓ , if for every ε > δ > x ∈ A where 0 < d X ( x, c ) < δ it follows that d Y ( f ( x ) , f ( c )) < ε. In other words, lim x → c f ( x ) = ℓ if for every ε > δ > x ∈ ( V δ ( c ) ∩ A ) \ { c } implies f ( x ) ∈ V ε ( ℓ ).Definition 6.3 is a far cry from the intuition students develop in a calculus classwhen it comes to discussing the definition of a limit of a real valued function atsome real number c . Calculus textbooks often use a statement such as “the limitof a function at c is the value the outputs approach as the inputs approach c ”and related figures to help the reader understand the situation. However, suchstatements leave a lot to be desired. They are neither technically robust nor arethe associated figures clearly connected to a technical definition such as Definition6.3. There is room for improvement.An alternative stems from concept of the accretion of a function associated witha point. The idea is to map neighborhoods of the point in the domain first, beforeconcerning ourselves ε -neighborhoods of the output of the point. After all, incalculus classes our students may be taught to develop an intuition about a functionby plugging values in and seeing what happens with the outputs. Definition 6.4. Let ( X, d X ) and ( Y, d Y ) be metric spaces, let B, D ⊆ X , and let f : D → Y . Given a set point c ∈ X , the accretion of f at c with respect to B ,denoted by A ( f, c, B ), is the set given by A ( f, c, B ) = \ δ> f ( V δ ( c ) ∩ B ) . (6.1)If D ∩ B = ∅ or if V δ ( c ) ∩ B = ∅ for some δ > 0, then A ( f, c, B ) = ∅ .When first encountered, Definition 6.4 may seem to be just as opaque as theclassical definition of limit of function provided by Definition 6.3. However, Defini-tion 6.4 can be unpacked and connected to the intuition students (might) developin calculus as follows: By definition, not all limits are limit points. Consider a constant sequence compared to thesingleton comprising the constant. RBITRARILY CLOSE 15 (1) First, for each δ > 0, the set of points of interest are gathered into the set V δ ( c ) ∩ B (the points in B within δ of c ).(2) Following intuition developed in calculus (maybe), the function is thenevaluated at these points.(3) For the next step, which is key to the development from the material fromthis point on, the closure of the image f ( V δ ( c ) ∩ B )—that is, the set ofpoints in the codomain arbitrarily close to this image—is taken for each δ > δ , the final step is to intersectthese closures over all positive δ , solidifying the intuitive notion of having “ δ approaches 0”. Taking this intersection amounts to the intuition of keepingonly points arbitrarily close to the outputs that always remain as the inputsapproach c .(5) The resulting set A ( f, c, B ), the accretion of f at c with respect to B , isthus the set of points the outputs of f approach as the inputs approach c .An immediate consequence of the definition of accretion for functions (Definition6.4) and follows from the fact that in a metric space, intersections of closed sets areclosed (Lemma 4.2). Corollary 6.5. Let ( X, d X ) and ( Y, d Y ) be metric spaces, let B, D ⊆ X , and let f : D → Y . Given any point c ∈ X and any set B ⊆ X , the set A ( f, c, B ) is closedin Y . For the sake of exposition, the examples and results obtained in the remainderof this section are limited to functions to and from the real line.The following examples illustrate the concept of accretion for functions and showthat care must be taken when comparing the definition of accretion in Definition6.4 with limits and continuity. Example 6.6. Consider Dirichlet’s function, the indicator function of the rationals Q : R → R given by Q ( x ) = ( , if x ∈ Q , , if x ∈ R \ Q . (6.2)Dirichlet’s function is a classic example of a nowhere-continuous function as dis-cussed in Remark 7.4 below. Due the density of the rational numbers Q and irra-tional numbers R \ Q in the real line R (that is, the fact that every δ -neighborhoodof any real number contains both rational and irrational numbers, see [1, Theorems1.4.3 and 1.4.4, p.22]), for each c ∈ R it follows that A ( Q , c, R ) = { , } . (6.3)That is, the accretion of Q at any real number c with respect to the domain R isthe set containing 0 and 1. Example 6.7. Consider Thomae’s function t : R → R given by t ( x ) = , if x = 0 , /n, if x ∈ Q , x = m/n is in reduced form with n > , , if x ∈ R \ Q . (6.4) As discussed in [1, p.114], the function t is continuous on R \ Q and discontinuousat each rational number. Also, A ( t, c, R ) = { , } , if c = 0 , { , /n } , if c ∈ Q , c = m/n is in reduced form with n > , { } , if c ∈ R \ Q . (6.5) Example 6.8. Consider the function f : R → R given by f ( x ) = ( , if x = 0 , /x, if x = 0 . (6.6)This function is continuous everywhere in R except at c = 0 where it is locallyunbounded and the limit does not exist. That is, at c = 0 and for each δ > ∈ f ( V δ (0) ∩ R ) and, moreover, f ( V δ (0) ∩ R ) = ( −∞ , − /δ ] ∪ { } ∪ [1 /δ, ∞ ) . (6.7)The intersection of these closures have only 0 in common, so A ( f, , R ) = \ δ> f ( V δ (0) ∩ R ) = { } . (6.8)Thus, even though the accretion of f at c = 0 with respect to R is a singleton,lim x → f ( x ) does not exist and f is not continuous at c .To avoid issues with existence of limits as in the previous example, the localbehavior of the function needs to be constrained somewhat. Definition 6.9. Let D ⊆ R , let f : D → R , and let c be an accumulation point of D . The function f is said to be locally bounded at c if f ( D ∩ V δ ( c )) is bounded forevery δ > Definition 6.10. Let D ⊆ R , let f : D → R , and let c be an accumulation pointof D . The accretion limit of f at c , denoted by alim x → c f ( x ), is said to exist if(i) f is locally bounded at c , and(ii) A ( f, c, D \ { c } ) is a singleton.In this case, A ( f, c, D \ { c } ) = { alim x → c f ( x ) } . If c is not an accumulation point of D , then f is said to not have an accretion limit at c .Note that the value of f at c is irrelevant in the context of accretion limits at c and, formally, it follows A ( f, c, D \ { c } ) = \ δ> f ( D ∩ V δ ( c ) \ { c } )(6.9) = \ δ> { f ( x ) : x ∈ D and 0 < | x − c | < δ } . (6.10)Aligning with intuition, when the accretion limit of f at c exists and as δ > f ( D ∩ V δ ( c ) \ { c } ) shrink down tothe singleton containing just alim x → c f ( x ). Ultimately, limits and accretion limitscoincide. The following lemma provides a step towards a proof of this statement. Lemma 6.11. If f : D → R , D ⊆ R , c is an accumulation point of D , and f islocally bounded at c , then the accretion A ( f, c, D \ { c } ) is a nonempty compact set. RBITRARILY CLOSE 17 Proof. Assume all of the hypotheses hold. By Lemma 6.5, A ( f, c, D \ { c } ) is closed.Since f is locally bounded at c , A ( f, c, D \ { c } ) is bounded. So by the Heine-BorelTheorem [1, Theorem 3.3.8, p.98], A ( f, c, D \ { c } ) is compact.Let ( x n ) be a sequeunce of points in D \ { c } where lim x n = c . It follows that( f ( x n )) is a bounded set of real numbers. So by the Bolzano-Weierstrass Theorem[1, Theorem 2.5.5, p.64], there is a real number y and a subsequence ( x n k ) of ( x n )such that lim k →∞ f ( x n k ) = y . Since lim k →∞ x n k = c (see [1, Theorem 2.5.2,p.63]), it follows that for every δ > K ( δ ) such thatfor all k ≥ K ( δ ) it must be that x n k ∈ V δ ( c ) ∩ D \ { c } . Since lim k →∞ f ( x n k ) = y implies y ∈ A (( f ( x n k ))) by Proposition 5.7, it follows that y acl f ( V δ ( c ) ∩ D \ { c } )for every δ > 0. Therefore, y ∈ A ( f, c, D \ { c } ). (cid:3) The following result, which is the last one in this section, connects various equiv-alent statements regarding functional limits. Theorem 6.12. Let D ⊆ R , let f : D → R , and let c be an accumulation point of D . Then the following statements are equivalent: (i) lim x → c f ( x ) = ℓ . (ii) For any sequence ( x n ) of points in D where lim n →∞ x n = c and x n = c forall n ∈ N , it follows that lim n →∞ f ( x n ) = ℓ . (iii) alim x → c f ( x ) = ℓ .Proof. The equivalence of (i) and (ii) is well known. See [1, Theorem 4.2.3, p.118],for instance. Thus, it suffices to show (ii) and (iii) are equivalent.Throughout this proof, fix a real number ℓ .To show (ii) implies (iii), first suppose f is not locally bounded at c . Then thereis a sequence ( z n ) of points in D \ { c } such that lim n →∞ z n = c but | f ( z n ) | > ℓ + 1for large enough n . Then lim n →∞ f ( z n ) = ℓ .Now suppose f is locally bounded at c but A ( f, c, D \ { c } ) = { ℓ } . By Lemma6.11, A ( f, c, D \ { c } ) is not empty, so there is some y ∈ A ( f, c, D \ { c } ) where ℓ = y .As in the proof of Lemma 6.11, there is a sequence ( y k ) of points in D \ { c } suchthat lim k →∞ y k = c and lim k →∞ f ( y k ) = y = ℓ .To show (iii) implies (ii), suppose there is a sequence ( x n ) of points in D \ { c } such that lim n →∞ x n = c but lim n →∞ f ( x n ) = ℓ . If ( f ( x n )) is unbounded, then f is not locally bounded at c . Therefore alim x → c f ( x ) = ℓ .Now suppose ( f ( x n )) is bounded and there is a sequence ( x n ) of points in D \ { c } such that lim n →∞ x n = c but lim n →∞ f ( x n ) = ℓ . By the Bolzano-WeierstrassTheorem [1, Theorem 2.5.5, p.64], there is a real number y and a subsequence ( x n k )of ( x n ) such that lim k →∞ f ( x n k ) = y where y = ℓ . Then for every δ > K ( δ ) such that for all k ≥ K ( δ ) it follows that x n k ∈ V δ ( c ) ∩ D \{ c } .Hence, y acl f ( V δ ( c ) ∩ D \ { c } ) for every δ > 0. Therefore, y ∈ A ( f, c, D \ { c } ) andso A ( f, c, D \ { c } ) = { ℓ } . Thus alim x → c f ( x ) = ℓ . (cid:3) The next section connects the notions of arbitrarily close and accretion of func-tions to the classical notions of continuity and differentiation.7. ACCRETION, CONTINUITY, AND DIFFERENTIATION As in the second half of the previous section, the examples and results obtainedin this remainder of this section are limited to functions to and from the real line R with its standard metric for the sake of exposition. The results obtained so far readily yield some connections to continuity, derivatives, and integrals of real valuedfunctions on the real line.First, thanks to Theorem 6.12 and the deep classical connection between func-tional limits and pointwise continuity, the next result readily follows. Note that c is assumed to be both an accumulation point and an element of the domain. Theorem 7.1. Let D ⊆ R , let f : D → R , and let c be in an accumulation pointof D that is also in D . Then the following statements are equivalent: (i) lim x → c f ( x ) = f ( c ) . (ii) For any sequence ( x n ) of points in D where lim n →∞ x n = c and x n = c forall n ∈ N , it follows that lim n →∞ f ( x n ) = f ( c ) . (iii) alim x → c f ( x ) = f ( c ) .Proof. Statements (i) and (ii) are equivalent to the classical definition of pointwisecontinuity for real valued function on the real line as found in [1, Definition 4.3.1,p.122] and proven with [1, Theorem 4.3.2, p.123]. The equivalence of statement(iii) with both (i) and (ii) follows from Theorem 6.12. (cid:3) The equivalence of statement (iii) with classical notions of continuity in a mildlyweakened setting motivate the following definition. Definition 7.2. Let D ⊆ R , let f : D → R , and let c be an accumulation point of D that is also in D . Then f is said to be accrete-continuous at c if f ( c ) = alim x → c f ( x ) . (7.1)The next corollary follows readily. Corollary 7.3. Let D ⊆ R , let f : D → R , and let c be an accumulation point of D that is also in D . Then f is accrete-continuous at c if and only if f is locallybounded at c and A ( f, c, D ) = { f ( c ) } .Proof. The result follows immediately from Definitions 6.10 and 7.2. (cid:3) Thus a real valued function on a set of real numbers is continuous at c if andonly if f is locally bounded at c and the accretion of f at c with respect to thedomain is the singleton comprising f ( c ). So, following intuition, as δ > f become closer and closerto f ( c ) and no other value. Remark . Corollary 7.3 tells us that Dirichlet’s function from Example 6.6 isnot continous anywhere on R since its accretion at any point contains two points.Similarly, Thomae’s function from Example 6.7 is continuous at each irrationalnumber (where the accretion contains only 0) and discontinuous at each rationalnumber (where the accretion contains two points).To see how the notion of accretion limit ties into differentiation, recall the defi-nition of derivative for real valued functions on the real line. Definition 7.5. Let I ⊆ R be an interval, let g : I → R , and let c ∈ I . Then the derivative of g at c is defined by g ′ ( c ) = lim x → c g ( x ) − g ( c ) x − c , (7.2)provided this limit exists. In this case, g is said to be differentiable at c . RBITRARILY CLOSE 19 Thus, if a real valued function g defined on an interval I has a derivative at c , then by Theorem 6.12 and Definition 6.10 it follows that g ′ ( c ) is the only realnumber arbitrarily close to the set of difference quotients that pass through thepoint ( c, g ( c )) restricted to the deleted δ -neighborhhood of c for any given δ > x gets closer and closer to c (hence δ istaken to be smaller and smaller), the slopes of the secant lines get closer and closerto the slope of the tangent line. Also, the following definition is now motivated. Definition 7.6. Let I ⊆ R be an interval, let g : I → R , and let c ∈ I . Thefunction g is said to be accrete-differentiable at c if g a ( c ) = alim x → c g ( x ) − g ( c ) x − c , (7.3)provided this accretion limit exists. In this case, g a ( c ) is called the accretion deriv-ative of g at c .When the accretion derivative g a ( c ) exists, it follows that { g a ( c ) } = \ δ> (cid:26) g ( x ) − g ( c ) x − c : x ∈ V δ ( c ) ∩ ( I \ { c } ) (cid:27) = A ( q, c, I \ { c } )(7.4)where q ( x ) is the difference quotient of g at c is defined for all x in I \ { c } by q ( x ) = g ( x ) − g ( c ) x − c . (7.5)Also, a function is differentiable at a point if and only if it is accretion-differentiableat the point. Corollary 7.7. Let I ⊆ R be an interval, let g : I → R , and let c ∈ I . Then g isdifferentiable at c if and only if g is accrete-differentiable at c .Proof. The result follows immediately from Theorem 6.12 along with Definitions7.5 and 7.6. (cid:3) An version of the Interior Extremum Theorem follows immediately from theDefinition 7.6 and the idea that 0 is the only real number arbitrarily close to thesets of positive and negative numbers. See [1, Theorem 5.2.6, p.151] for comparison. Theorem 7.8. Suppose f : ( a, b ) → R , f is accrete-differentiable at every point inthe open interval ( a, b ) , and f attains its maximum at some point c in ( a, b ) . Then f a ( c ) = 0 .Proof. Let δ > 0. Suppose x, y ∈ ( a, b ) with c − δ < x < c < y < c + δ . Then x − c < < y − c, f ( x ) − f ( c ) ≤ , and f ( y ) − f ( c ) ≤ . (7.6)Hence f ( y ) − f ( c ) y − c ≤ ≤ f ( x ) − f ( c ) x − c . (7.7)Therefore, f a ( c ) is arbitrarily close to both the set of nonnegative real numbersand the set of nonpositive real numbers. By Lemma 2.6, f a ( c ) = 0. (cid:3) INTEGRATION AND THE EVALUATION PART OF THEFUNDAMENTAL THEOREM OF CALCULUS Following the content in Section 2, a bit more notation and terminology actu-ally provides enough material with to define an integral of a bounded real valuedfunction f on a closed and bounded interval [ a, b ]. Definition 8.1. A partition of a closed and bounded interval [ a, b ] is a finite set P of points of [ a, b ] such that P = { x = a, x , . . . , x n = b } ⊆ R n where x < x < . . . < x n . (8.1)A partition of [ a, b ] is also said to partition [ a, b ]. A weight is a vector w ∈ S m ∈ N R m .For a partition of [ a, b ] given by P = { x , . . . , x n } and any weight v = ( v , . . . , v n )in R n , the weighted sum s ( P, v ) is defined by s ( P, v ) = n X k =1 v k ( x k − x k − ) . (8.2)Weighted sums allow one to literally think outside the box. Given a boundedfunction on a compact interval, the endpoints of this compact interval along withthe supremum and infimum of the range of the function form a box in which thegraph of the function resides. The idea is to approximate the integral in question bychoosing values to represent the height of the function which lie outside of this box,then refining the approximation with finer and finer partitions, thus finer and finerboxes. The result is an alternative definition of integration that relies on neitherRiemann sums nor Darboux sums. Definition 8.2. Let f : [ a, b ] → R and suppose f is bounded. Let P be a partitionof [ a, b ] given by P = { x o , . . . , x n } with x < x < . . . < x n . The set of upperweights of f with respect to P is defined by U ( f, P ) = { a = ( a , . . . , a n ) ∈ R n : a k ≥ f ( x ) for x ∈ [ x k − , x k ] , k = 1 , . . . , n } . (8.3)Similarly, the set of lower weights of f with respect to P is defined by L ( f, P ) = { b = ( b , . . . , b n ) ∈ R n : b k ≤ f ( x ) for x ∈ [ x k − , x k ] , k = 1 , . . . , n } . (8.4)Lastly, the set of upper sums of f is defined by U f = { s ( P, a ) : P is a partition of [ a, b ] , a ∈ U ( f, P ) } , (8.5)and the set of lower sums of f is defined by L f = { s ( P, a ) : P is a partition of [ a, b ] , b ∈ L ( f, P ) } . (8.6)Note that U f and L f are sets of real numbers while U ( f, P ) and L ( f, P ) are setsof vectors.In Definition 8.3, real number arbitrarily close to both L f and U f is the integral of f . Moreover, the integral defined in this way makes use of neither tags norsuprema and infima. Thus, Definition 8.3 is distinct from the Darboux and Riemannintegrals. Still, these notions of integration are equivalent, as shown in Theorem8.8. RBITRARILY CLOSE 21 Definition 8.3. A bounded function f : [ a, b ] → R is said to be integrable over [ a, b ] if there is some y ∈ R such that y is arbitrarily close to both U f and L f . Inthis case, y is denoted by R ba f and called the integral of f over [ a, b ].After developing a couple of basic results in this framework, a version of theevaluation part of the Fundamental Theorem of Calculus can be proven. (SeeTheorem 8.7 below.)First, the following lemma establishes the fact that any upper sum of a givenfunction is greater than or equal to any lower sum of the same function, regardlessof the choice of partitions. Lemma 8.4. For a bounded function f : [ a, b ] → R , any pair of partitions P and P of [ a, b ] , and any choice of an upper weight a ∈ U ( f, P ) and a lower weight b ∈ L ( f, P ) , it follows that s ( P , b ) ≤ s ( P , a ) .Proof. Let P = { z , . . . , z n } and P = { y , . . . , y n } . Define P = P ∪ P andrewrite P as P = { x , x , . . . , x n } such that x < x < . . . < x n . (8.7)Given a = ( a , . . . , a n ) ∈ U ( f, P ) and b = ( b , . . . , a n ) ∈ L ( f, P ) , define a ′ = ( a ′ , . . . , a ′ n ) ∈ U ( f, P )(8.8)by a ′ k = a h when [ x k − , x k ] ⊆ [ z h − , z h ] for k = 1 , . . . , n and h = 1 , . . . , n , accord-ingly. Likewise, define b ′ = ( b ′ , . . . , b ′ n ) ∈ L ( f, P )(8.9)by b ′ k = b j when [ x k − , x k ] ⊆ [ y j − , y j ] for k = 1 , . . . , n and j = 1 , . . . , n , accord-ingly. Since b ′ k ≤ f ( x ) ≤ a ′ k for each k = 1 , . . . , n and any x ∈ [ x k − , x k ], partialtelescoping yields s ( P , b ) = s ( P, b ′ ) ≤ n X k =1 f ( x )( x k − x k − ) ≤ s ( P, a ′ ) = s ( P , a ) . (8.10) (cid:3) Lemma 8.4 leads immediately to the following result. Lemma 8.5. The integral of f over [ a, b ] is unique.Proof. By way of contradiction, suppose y < y but y acl U f , y acl L f , y acl U f ,and y acl L f . Choose ε = | y − y | / > 0. Then there are partitions P and P of [ a, b ] along with an upper weight a ∈ U ( f, P ) and a lower weight b ∈ L ( f, P )such that | s ( P , a ) − y | < ε and | s ( P , b ) − y | < ε . (8.11)Hence − ε < s ( P , a ) − y < ε and − ε < s ( P , b ) − y < ε . Since y + ε = 2 y + y < y + 2 y y − ε , (8.12)it follows that s ( P , a ) < y + ε < y − ε < s ( P , b ) . (8.13)These inequalities contradict Lemma 8.4 which says s ( P , b ) ≤ s ( P , a ) . Therefore, R ba f is unique. (cid:3) The Mean Value Theorem provides a key piece of the puzzle. Its statement isprovided here for convenience, but see [1, Theorem 5.3.2, p.156] for a proof. Theorem 8.6 (Mean Value Theorem) . If f : [ a, b ] → R is continuous on [ a, b ] anddifferentiable on ( a, b ) , then there is a point c ∈ ( a, b ) where f ′ ( c ) = f ( b ) − f ( a ) b − a . (8.14)A version of the evaluation half of the Fundamental Theorem of Calculus nowfollows readily from the definition of an integral given in Definition 8.3 and theresults in this section. In particular, neither Riemann sums nor Darboux sums areused in the proof. Theorem 8.7 (Evaluation half of the Fundamental Theorem of Calculus) . Suppose f and F are real valued functions on [ a, b ] where f is the derivative of F on [ a, b ] and f is integrable over [ a, b ] . Then R ba f = F ( b ) − F ( a ) .Proof. Let P be a partition of [ a, b ] with P = { x , . . . , x n } . By the Mean ValueTheorem 8.6, for each k = 1 , . . . , n there exists t k ∈ ( x k − , x k ) such that F ( x k ) − F ( x k − ) = F ′ ( t k )( x k − x k − ) = f ( t k )( x k − x k − ) . (8.15)Since f is integrable, it is bounded on [ a, b ]. By Lemma 8.4, any choice of weights a ∈ U ( f, P ) and b ∈ L ( f, P ) as well as any index k = 1 , . . . , n yields b k ≤ f ( t k ) ≤ a k and x k − x k − > . (8.16)Therefore, s ( P, b ) ≤ n X k =1 f ( t k )( x k − x k − ) = F ( b ) − F ( a ) ≤ s ( P, a ) . (8.17)Now, since f is integrable over [ a, b ], for every ε > P and P of [ a, b ] along with an upper weight a ∈ U ( f, P ) and a lower weight b ∈ L ( f, P ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba f − s ( P , a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba f − s ( P , b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. (8.18)By Lemma 8.4 and inequalities (8.18), s ( P , b ) − ε ≤ F ( b ) − F ( a ) − ε ≤ s ( P , a ) − ε < Z ba f (8.19)and Z ba f < s ( P , b ) + ε ≤ F ( b ) − F ( a ) + ε ≤ s ( P , a ) + ε. (8.20)Therefore, for every ε > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba f − ( F ( b ) − F ( a )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. (8.21)By Lemma 2.5, Z ba f = F ( b ) − F ( a ) . (8.22) (cid:3) RBITRARILY CLOSE 23 The proof of this version of the evaluation half of the Fundamental Theorem ofCalculus (Theorem 8.7) used neither the tags associated Riemann sums and nor thesuprema and infima associated with Darboux sums. Still, the three correspondingnotions of integration are equivalent. Theorem 8.8. Let f be a bounded real valued function on [ a, b ] . The following areequivalent :(i) f is integrable over [ a, b ] as defined in Definition 8.3. (ii) f is Darboux-integrable over [ a, b ] ([1, Definition 7.2.7, p.220]) . (iii) f is Riemann-integrable over [ a, b ] ([1, Theorem 8.1.2, p.251]) . The equivalence of (ii) and (iii) is a classical result in analysis. See [1, Theorem8.1.2, p.251] for details. To explore the possible equivalence of (i) with (ii) and (iii),but in lieu of fully developing Darboux and Riemann integrals on their own, considerthe various sums used in each of the definitions of integrability, respectively.To establish the connection to Riemann sums, let P = { x , . . . , x n } be a partitionof [ a, b ]. Consider a choice of a vector c = ( c , . . . , c n ) ∈ R n where c k ∈ [ x k − , x k ]for each k = 1 , . . . , n , define f ( c ) = ( f ( c ) , . . . , f ( c n )) . (8.23)Then the classical Riemann sum (see [1, p.250]) with ‘tags’ c = ( c , . . . , c n ) ∈ R n is given by the weighted sum with weights given by f ( c ) as follows: s ( P, f ( c )) = n X k =1 f ( c k )( x k − x k − ) . (8.24)As a result, for any partition P of [ a, b ] and any choice of upper weights a ∈ U ( f, P )and lower weights b ∈ L ( f, P ), it follows that since b k ≤ f ( x ) ≤ a k for each k = 1 , . . . , n and any x ∈ [ x k − , x k ] s ( P , b ) ≤ n X k =1 f ( x )( x k − x k − ) ≤ s ( P , a ) . (8.25)Now, to establish the connection to Darboux sums, again let P = { x , . . . , x n } be a partition of [ a, b ]. Define u = ( u , . . . , u n ) ∈ U ( f, P ) by u k = sup { f ( x ) : x ∈ [ x k − , x k ] } (8.26)and define w = ( w , . . . , w n ) ∈ L ( f, P ) by w k = inf { f ( x ) : x ∈ [ x k − , x k ] } , (8.27)Then the classical upper and lower Darboux sums (see [1, Definition 7.2.1, p.218])are given by the weighted sums weighted by u and w , respectively. That is, s ( P, u ) = n X k =1 u k ( x k − x k − ) and s ( P, w ) = n X k =1 w k ( x k − x k − ) . (8.28)So for any a ∈ U ( f, P ) and b ∈ L ( f, P ) it follows that s ( P, b ) ≤ s ( P, w ) ≤ s ( P, f ( c )) ≤ s ( P, u ) ≤ s ( P, a ) . (8.29)To complete the proof of Theorem 8.8 it suffice to show (i) is equivalent to (ii).To this end, the following equivalent form of the Darboux integral [1, Theorem7.2.8, p.221]—which itself is reminiscent of the definition of arbitrarily close—isquite useful. Theorem 8.9. A bounded function f is integrable on [ a, b ] if and only if, for every ε > , there exists a partition P ε of [ a, b ] such that s ( P, u ) − s ( P, w ) < ε, (8.30) where u and w are as in (8.26) and (8.27) , respectively.Proof of Theorem 8.8. Assume f is Darboux-integrable over [ a, b ]. Let R ba f denotethe Darboux integral of f , which by [1, Definition 7.2.7, p.220] yields Z ba f = inf { s ( P, u ) : upper Darboux sum s ( P, u ) , P partitions [ a, b ] } (8.31)and Z ba f = sup { s ( P, w ) : lower Darboux sum s ( P, w ) , P partitions [ a, b ] } . (8.32)Since s ( P, u ) ∈ U f and s ( P, w ) ∈ L f for any corresponding weights u and w definedas in (8.26) and (8.27), respectively, it follows from Definition 3.2 that R ba f acl U f and R ba f acl L f . Therefore, f is integrable over [ a, b ] in the sense of Definition 8.3.Now suppose f is integrable over [ a, b ] in the sense of Definition 8.3 where R ba f acl U f and R ba f acl L f . Let ε > 0. Then there is a partition P of [ a, b ] alongwith an upper weight a ∈ U ( f, P ) and a lower weight b ∈ L ( f, P ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba f − s ( P, a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba f − s ( P, b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε/ . (8.33)Let weights u and w be defined as in (8.26) and (8.27), respectively. Then s ( P, b ) ≤ s ( P, w ) ≤ s ( P, u ) ≤ s ( P, a )(8.34)from which it follows that − s ( P, a ) ≤ − s ( P, u ) ≤ − s ( P, w ) ≤ − s ( P, b ) . (8.35)A combination of inequalities (8.33), (8.34), and (8.35) yields − ε/ < s ( P, b ) − Z ba f ≤ s ( P, u ) − Z ba f ≤ s ( P, a ) − Z ba f < ε/ ε/ < Z ba f − s ( P, a ) ≤ Z ba f − s ( P, w ) ≤ Z ba f − s ( P, b ) < ε/ . (8.37)Therefore, s ( P, u ) − s ( P, w ) < ε. (8.38)By Theorem 8.9, f is Darboux-integrable over [ a, b ]. (cid:3) The paper closes with some thoughts on pedagogy and some exercises. RBITRARILY CLOSE 25 CLOSURE: PEDAGOGY AND EXERCISES A quantifier-heavy way to write out the definition of arbitrarily close, Definition2.2, in the context of the real line is as follows: ℓ acl S ⇐⇒ ∀ ε > , ∃ x ∈ S such that | x − ℓ | < ε. (9.1)For the sake of comparison, consider the definition of the limit of a sequence (Def-inition 2.8) stated with quantifiers:lim n →∞ a n = ℓ ⇐⇒ ∀ ε > , ∃ N ∈ N such that ∀ n ≥ N, | a n − ℓ | < ε. (9.2)Statement (9.2) is difficult for students to understand at first, which is why oneshould consider using a technical definition of arbitrarily close such as Definition2.2 as a bridge to limits and convergence.Note that only two qualifiers are used in the definition of arbitrarily close. Forthis reason, discussions of arbitrarily close could reasonably be part of any coursethat serves as an introduction to writing mathematical proofs.Actually, there are topics in undergraduate real analysis that could reasonablybe included in a proofs-based course. For instance, writing logical statements inthe manner just above is not always the best way to understand them, but doingso allows for negations to be readily stated. In the case where a real number y isnot arbitrarily close to a set of real number B , one could write ∃ ε > ∀ x ∈ B it follows that | x − y | ≥ ε . (9.3)Thus, the negation of arbitrarily close leads to a statement that is essentiallythe property that defines what it means for a set to be open in the topology of thereal line, as discussed in the dialog leading to Definition 4.3. Look at the statementagain. It immediately implies that each real number y that is not arbitrarily closeto B lies in an open interval that does not intersect B .Also consider the case when a set has an upper bound. Such a set is said tobe bounded above . See Definition 3.1 above as well as [1, Definition 1.3.1, p.15].The intuition students have by the time they reach a first course in analysis servesthem well when it comes to dealing with bounded and unbounded sets. In termsof quantifiers, a set S is said to be bounded when ∃ b > ∀ x ∈ S it follows that | x | ≤ b. (9.4)Of course, if S is unbounded , it follows that ∀ b > , ∃ x ∈ S such that | x | > b. (9.5)A simple intuitive idea is that quantified statements of the form ∃ . . . followedby ∀ . . . seem to indicate the existence of a constraint of some kind. That is,whatever exists dictates something about everything in some collection of objects.On the other hand, statements of the form ∀ . . . followed by ∃ . . . definitely indicatethe existence of a function. In particular, the first pairing of quantifiers in thedefinition of the limit of sequence (Definition 2.8) given by ∀ ε > , ∃ N ∈ N . . . (9.6)is often taken to mean that N is a function of the input given by the error ε > ∃ N ∈ N such that ∀ n ≥ N . . . (9.7) can be taken to mean the N provides a constraint on all of the indeces n that follow.Therefore, N is a function of ε > n which indicateshow far into the sequence one needs to go to ensure the terms are as close to thelimit as desired.Ultimately, an important question is as follows: Where should an introductorycourse on real analysis begin? One might consider kicking things off with a for-mal definition of something many mathematicians seem to intuitively understand:What it means for things to be arbitrarily close. This can be done even beforethe definition of supremum and certainly before any notion of convergence or limitis discussed. The time spent on such a definition could serve as an intermediatestep towards an understanding of the subtleties of convergence and other abstractmathematical concepts.Many of the results proven throughout the paper would serve as an interestingexercise in a course on undergraduate analysis. The paper concludes with additionalexercises for the reader to try. It is suggested that solutions to these exercises bestated in terms of the technical definition of arbitrarily close along with the variousnotions of accretion presented in this paper. Exercise 9.1. Let a and b be two points in a metric space. Prove a = b if andonly if a acl { b } . Exercise 9.2. Find the closure as well as the complement of the closure of each ofthe following subsets of the real line. Draw stuff. a. A = [0 , b. B = (0 , ∪ { } c. C = N d. D = { − ( − n /n : n ∈ N } e. E = { − n + 1 /n : n ∈ N } f. F = { . , . , . , . . . } Exercise 9.3. Find an example of sequence of real numbers for which its accretion A (( x n )) is bounded and countable. Exercise 9.4. Prove that for every x ∈ R it follows that x acl Q (thus Q = R ). Exercise 9.5. Prove that the interval (0 , π ] is neither open nor closed. Exercise 9.6. Consider the interval I = [3140 , n ∈ N , can some y n ∈ I such that | y n − | < / n always be found? Exercise 9.7. A sequence of real numbers ( x n ) is a said to be strictly increasing if x n < x n +1 for each n ∈ N . Prove that if sup S exists but sup S / ∈ S , then there is astrictly increasing sequence ( x n ) of points in S such that for each n ∈ N it followsthat | x n − sup S | < /n . Exercise 9.8. Prove that lim n →∞ x n = ℓ if and only if the range of every subse-quence of ( x n ) is arbitrarily close to ℓ . Exercise 9.9. Prove lim n →∞ (1 /n + 3140 + ( − n ) does not exist. Exercise 9.10. Prove lim x → | x − | x − does not exist. Exercise 9.11. Consider the function g : (0 , ∞ ) → R given by g ( x ) = 5 cos(1 /x ).Draw a figure for g . Also, find three sequences of positive numbers ( x n ) , ( y n ) , and( z n ) where lim x n = lim y n = lim z n = 0 but where lim g ( x n ) = 0 , lim g ( y n ) = 5 , and lim g ( z n ) = − 5. What can be said about lim x → g ( x )? For every δ > 0, whichreal numbers are arbitrarily close to the set g ((0 , δ )) = { g ( x ) : 0 < x < δ } ? RBITRARILY CLOSE 27 Exercise 9.12. Let g : R → R be defined by g ( x ) = x cos 1 x , if x = 0 , , if x = 0 . Prove g ′ (0) exists and compute g ′ ( x ) for x = 0. Determine which real numbers arearbitrarily close to g ′ ( V δ (0) \ { } ) for every δ > 0. Is g ′ integrable on [0 , Exercise 9.13. Prove that Dirichlet’s function from Example 6.6 is not integrableover [0 , Exercise 9.14. Prove that Thomae’s function from Example 6.7 is integrable over[0 , References 1. S. Abbott, Understanding Analysis , 2nd edition, Springer, New York, NY, 2015.2. C. Adams and R. Franzosa, Introduction to Topology: Pure and Applied , Pearson, NJ, 2008.3. J. Barnes (2007) Teaching Real Analysis in the Land of Make Believe , PRIMUS, vol. 14,Taylor and Francis, UK, 2007, pp. 366–372.4. J. E. Borzellino, WHOSE LIMIT IS IT ANYWAY? , PRIMUS, vol. 11, Taylor and Francis,UK, 2001, pp. 265–274.5. B. Cornu, Limits , Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht,Netherlands, 1991, pp. 153–166.6. Seager, S. (2019) Analysis Boot Camp: An Alternative Path to Epsilon-Delta Proofs in RealAnalysis , PRIMUS, Taylor and Francis, UK, 2019 (to appear). Department of Mathematics and Statistics, Cal Poly Pomona, 3801 W Temple Ave,Pomona, CA 91768 E-mail address : [email protected] URL ::