Part I

140A Homework Exercises (Fall 2012)

3

• The problems below either come from the lecture notes or from Rudin’s “Principles of Real Analysis,”3rd ed.

• Although pictures are not typically provided in these solutions, I would highly advise that the readerdraw appropriate pictures whenever possible. Many of the proofs to follow are really a transcription of avery natural picture proof to a formal written proof.

1

HomeWork #1 Due Thursday October 4, 2012

Exercise 1.1. Show that all convergent sequences {an}∞n=1 ⊂ Q are Cauchy.

Exercise 1.2. Show all Cauchy sequences {an}∞n=1 are bounded – i.e. there exists M ∈ N such that

|an| ≤M for all n ∈ N.

Exercise 1.3. Suppose {an}∞n=1 and {bn}∞n=1 are Cauchy sequences in Q. Show {an + bn}∞n=1 and{an · bn}∞n=1 are Cauchy.

Exercise 1.4. Assume that {an}∞n=1 and {bn}∞n=1 are convergent sequences in Q. Show {an + bn}∞n=1 and{an · bn}∞n=1 are convergent in Q and

limn→∞

(an + bn) = limn→∞

an + limn→∞

bn and

limn→∞

(anbn) = limn→∞

an · limn→∞

bn.

Exercise 1.5. Assume that {an}∞n=1 and {bn}∞n=1 are convergent sequences in Q such that an ≤ bn for alln. Show A := limn→∞ an ≤ limn→∞ bn =: B.

Exercise 1.6 (Sandwich Theorem). Assume that {an}∞n=1 and {bn}∞n=1 are convergent sequences in Qsuch that limn→∞ an = limn→∞ bn. If {xn}∞n=1 is another sequence in Q which satisfies an ≤ xn ≤ bn for alln, then

limn→∞

xn = a := limn→∞

an = limn→∞

bn.

Please note that that main part of the problem is to show that limn→∞ xn exists in Q. Hint: start byshowing; if a ≤ x ≤ b then |x| ≤ max (|a| , |b|) .

Exercise 1.7. Use the following outline to construct another Cauchy sequence {qn}∞n=1 ⊂ Q which is notconvergent in Q.

1. Recall that there is no element q ∈ Q such that q2 = 2. To each n ∈ N let mn ∈ N be chosen so that

m2n

n2< 2 <

(mn + 1)2

n2(1.1)

and let qn := mn

n .2. Verify that q2n → 2 as n→∞ and that {qn}∞n=1 is a Cauchy sequence in Q.3. Show {qn}∞n=1 does not have a limit in Q.

2

HomeWork #2 Due Thursday October 11, 2012

Problems from Rudin:Chapter 1: 1.1, 1.4.Chapter 3: 3.2Note: this problem may seem trivial, but the point is to realize that a least upper bound for E need not

actually be an element of E!

Definition 2.1 (Subsequence). We say a sequence, {yk}∞k=1 is a subsequence of another sequence,{xn}∞n=1 , provided there exists a strictly increasing function, N 3 k → nk ∈ N such that yk = xnk

forall k ∈ N. [Example, nk = k2 + 3, and {yk := xk2+3}∞k=1 would be a subsequence of {xn}∞n=1 .]

Exercise 2.1. Suppose that {xn}∞n=1 is a Cauchy sequence in Q (or R) which has a convergent subsequence,{yk = xnk

}∞k=1 in Q (or R). Show that limn→∞ xn exists and is equal to limk→∞ yk.

Exercise 2.2. Suppose that α ⊂ Q is a cut as in Definition 2.27. Show α is bounded from above. Then letm := supα and show that α = αm, where αm

αm := {y ∈ Q : y < m} .

Also verify that αm is a cut for all m ∈ R. [In this way we see that we may identify R with the cuts of Q.This should motivate Dedekind’s construction of the real numbers as described in Rudin.]

Exercise 2.3 (Do not hand in). Suppose that {an} and {bn} are sequences of real numbers such thatA := limn→∞ an and B := limn→∞ bn exists in R. Then;

1. {an}∞n=1 is a Cauchy sequence.2. limn→∞ |an| = |A| .3. If A 6= 0 then limn→∞

1an

= 1A .

4. limn→∞ (an + bn) = A+B.5. limn→∞ (anbn) = A ·B.6. If an ≤ bn for all n, then A ≤ B.7. If {xn}∞n=1 ⊂ R is another sequence such that an ≤ xn ≤ bn and A = B, then limn→∞ xn = A = B.

[The point here is to convince yourself that the proofs of the analogous statement you gave when ev-erything was in Q still hold true here. This would be a good time to make sure that you understand theseproofs!]

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HomeWork #3 Due Thursday October 17, 2012

Problems from Rudin:Chapter 1: 1.6 (Hint: for part d) use the Sup-Sup theorem or its corollary.)Chapter 3: 3.3, 3.4. (Look at but do not hand in. Hint: derive formulas for s2n and s2n+1.)

10 3 HomeWork #3 Due Thursday October 17, 2012

Exercise 3.1. Let {an}∞n=1 be the sequence given by,

(−1, 2, 3,−1, 2, 3,−1, 2, 3, . . . ) .

Find lim supn→∞ an and lim infn→∞ an.

Exercise 3.2. Show lim infn→∞(−an) = − lim supn→∞ an.

Exercise 3.3 (Do not hand in). If {an}∞n=1 and {bn}∞n=1 are two sequences such that an ≤ bn for a.a. n,then

lim supn→∞

an ≤ lim supn→∞

bn and lim infn→∞

an ≤ lim infn→∞

bn. (3.1)

Exercise 3.4. Suppose that {an}∞n=1 and {bn}∞n=1 are sequences in R. Show

lim supn→∞

(an + bn) ≤ lim supn→∞

an + lim supn→∞

bn (3.2)

provided that the right side of Eq. (3.8) is well defined, i.e. no ∞−∞ or −∞ +∞ type expressions. (It isOK to have ∞+∞ =∞ or −∞−∞ = −∞, etc.)

Exercise 3.5. Suppose that {an}∞n=1 and {bn}∞n=1 are sequences in [0,∞) ⊂ R. Show

lim supn→∞

(anbn) ≤ lim supn→∞

an · lim supn→∞

bn, (3.3)

provided the right hand side of (3.9) is not of the form 0 · ∞ or ∞ · 0.

3.1 Test #1, Monday 10/22/2012

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HomeWork #4 Due Thursday October 25, 2012

You should do these problems but they will not be collected this week.Problems from Rudin:Chapter 1: 1.11, 1.12, 1.13, 1.17 [This will appear on the next homework.]

Exercise 4.1. Let (F, P ) be an ordered field and x, y ∈ F with y > x. Show;

1. y + a > x+ a for all a ∈ F,2. −x > −y,3. if we further suppose x > 0, show 1

x >1y .

Exercise 4.2. Show limn→∞ αn =∞ and limn→∞1αn = 0 whenever α > 1.

Exercise 4.3. Suppose α > 1 and k ∈ N, show there is a constant c = c (α, k) > 0 such that αn ≥ cnk forall n ∈ N. [In words, αn grows in n faster than any polynomial in n.]

Exercise 4.4. Suppose that {an}∞n=1 is a sequence of real numbers and let

A := {y ∈ R : an ≥ y for a.a. n} .

Then supA = lim infn→∞ an with the convention that supA = −∞ if A = ∅.

Exercise 4.5. Suppose that {an}∞n=1 is a sequence of real numbers. Show lim supn→∞ an = a∗ ∈ R iff forall ε > 0,

an ≤ a∗ + ε for a.a. n. and

a∗ − ε ≤ an i.o. n.

Similarly, show lim infn→∞ an = a∗ ∈ R iff for all ε > 0,

an ≤ a∗ + ε i.o. n. and

a∗ − ε ≤ an for a.a. n.

Notice that this exercise gives another proof of item 2. of Proposition 3.28 in the case all limits are realvalued.

Exercise 4.6 (Cauchy Complete =⇒ L.U.B. Property). Suppose that R denotes any ordered fieldwhich is Cauchy complete and N is not bounded from above. Show R has the least upper bound propertyand therefore is the field of real numbers.

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HomeWork #5 Due Thursday November 1, 2012

Problems from Rudin:Chapter 1: 1.11, 1.12, 1.13, 1.17(1.17 is not to be handed in.)

Exercise 5.1. Let X be a set and for any function f : X → C, let

‖f‖u := supx∈X|f (x)| .

Show Z := {f : X → C : ‖f‖u <∞} is a vector space and ‖·‖u is a norm on Z.

For the next two exercise you will be using some concepts from calculus which we will developed in detailnext quarter. For now, I assume you know what the Riemann integral is for continuous functions on [0, 1]with values in R. Let Z denote the continuous functions1 on [0, 1] with values in R. The only properties thatyou need to know about the Riemann integral are;

1. The integral is linear, namely for all f, g ∈ Z and λ ∈ R,∫ 1

0

(f(t) + λg (t))dt =

∫ 1

0

f (t) dt+ λ

∫ 1

0

g (t) dt.

2. If f, g ∈ Z and f (t) ≤ g (t) for all t ∈ [0, 1] , then∫ 1

0

f (t) dt ≤∫ 1

0

g (t) dt.

3. For all f ∈ Z, ∣∣∣∣∫ 1

0

f (t) dt

∣∣∣∣ ≤ ∫ 1

0

|f (t)| dt.

In fact this item follows from items 1. and 2. Indeed, since ±f (t) ≤ |f (t)| for all t, we find

±∫ 1

0

f (t) dt =

∫ 1

0

±f (t) dt ≤∫ 1

0

|f (t)| dt ⇐⇒∣∣∣∣∫ 1

0

f (t) dt

∣∣∣∣ ≤ ∫ 1

0

|f (t)| dt.

Exercise 5.2. Let Z denote the continuous function on [0, 1] with values in R and for f ∈ Z let

‖f‖1 :=

∫ 1

0

|f (t)| dt.

Show ‖·‖1 satisfies,

1. (Homogeneity) ‖λf‖ = |λ| ‖f‖ for all λ ∈ R and f ∈ Z.1 The notion of continuity will be formally developed shortly.

16 5 HomeWork #5 Due Thursday November 1, 2012

2. (Triangle inequality) ‖f + g‖ ≤ ‖f‖+ ‖g‖ for all f, g ∈ Z.

Exercise 5.3. Let Z denote the continuous function on [0, 1] with values in R and for f ∈ Z let

‖f‖2 :=

√∫ 1

0

|f (t)|2 dt.

Show;

1. for f, g ∈ Z that ∣∣∣∣∫ 1

0

f (t) g (t) dt

∣∣∣∣ ≤ ‖f‖2 · ‖g‖2 ,2. Homogeneity) ‖λf‖2 = |λ| ‖f‖2 for all λ ∈ R and f ∈ Z, and3. (Triangle inequality) ‖f + g‖2 ≤ ‖f‖2 + ‖g‖2 for all f, g ∈ Z.

For the remaining exercises, let X and Y be sets, f : X → Y be a function and {Ai}i∈I be an indexedfamily of subsets of Y, verify the following assertions.

Exercise 5.4. (∩i∈IAi)c = ∪i∈IAci .

Exercise 5.5. Suppose that B ⊂ Y, show that B \ (∪i∈IAi) = ∩i∈I(B \Ai).

Exercise 5.6. f−1(∪i∈IAi) = ∪i∈If−1(Ai).

Exercise 5.7. f−1(∩i∈IAi) = ∩i∈If−1(Ai).

Exercise 5.8. Find a counterexample which shows that f(C ∩D) = f(C) ∩ f(D) need not hold.

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HomeWork #6 Due Thursday November 8, 2012

Exercise 6.1. Show that Qn is countable for all n ∈ N.

Exercise 6.2. Let Q [t] be the set of polynomial functions, p, such that p has rationale coefficients. That isp ∈ Q [t] iff there exists n ∈ N0 and ak ∈ Q for 0 ≤ k ≤ n such that

p (t) =

n∑k=0

aktk for all t ∈ R. (6.1)

Show Q [t] is a countable set.

Definition 6.1 (Algebraic Numbers). A real number, x ∈ R, is called algebraic number, if there is anon-zero polynomial p ∈ Q [t] such that p (x) = 0. [That is to say, x ∈ R is algebraic if it is the root of anon-zero polynomial with coefficients from Q.]

Note that for all q ∈ Q, p (t) := t − q satisfies p (q) = 0. Hence all rational numbers are algebraic. Butthere are many more algebraic numbers, for example y1/n is algebraic for all y ≥ 0 and n ∈ N.

Exercise 6.3. Show that the set of algebraic numbers is countable. [Hint: any polynomial of degree n hasat most n – real roots.] In particular, “most” irrational numbers are not algebraic numbers, i.e. there is stillan uncountable number of non-algebraic numbers..

Exercise 6.4. Show that convergent sequences in metric spaces are always Cauchy sequences. The converseis not always true. For example, let X = Q be the set of rational numbers and d(x, y) = |x − y|. Choosea sequence {xn}∞n=1 ⊂ Q which converges to

√2 ∈ R, then {xn}∞n=1 is (Q, d) – Cauchy but not (Q, d) –

convergent. Of course the sequence is convergent in R..

Exercise 6.5. If {xn}∞n=1 is a Cauchy sequence in a metric space (X, d), then limn→∞ d (xn, y) exists in Rfor all y ∈ X. In particular, {d (xn, y)}∞n=1 is a bounded sequence in R for all y ∈ X..

Exercise 6.6. Let X be a set and (Y, ρ) be a complete metric space. Suppose that fn : X → Y are functionssuch that

δm,n := supx∈X

d (fn (x) , fm (x))→ 0 as m,n→∞.

Show there exists a (unique) functions, f : X → Y such that

limn→∞

supx∈X

d (fn (x) , f (x)) = 0.

Hint: mimic the proof of Lemma 6.27.

Exercise 6.7. Suppose that (X, d) is a metric space and f, g : X → C are two continuous functions on X.Show;

1. f + g is continuous,

18 6 HomeWork #6 Due Thursday November 8, 2012

2. f · g is continuous,3. f/g is continuous provided g (x) 6= 0 for all x ∈ X.

Exercise 6.8. Show the following functions from C to C are continuous.

1. f (z) = c for all z ∈ C where c ∈ C is a constant.2. f (z) = |z| .3. f (z) = z and f (z) = z̄.4. f (z) = Re z and f (z) = Im z.

5. f (z) =∑Nm,n=0 am,nz

mz̄n where am,n ∈ C.

Exercise 6.9. Suppose now that (X, ρ), (Y, d), and (Z, δ) are three metric spaces and f : X → Y andg : Y → Z. Let x ∈ X and y = f (x) ∈ Y, show g ◦ f : X → Z is continuous at x if f is continuous at x andg is continuous at y. Recall that (g ◦ f) (x) := g (f (x)) for all x ∈ X. In particular this implies that if f iscontinuous on X and g is continuous on Y then f ◦ g is continuous on X.

Exercise 6.10 (Intermediate value theorem). Suppose that −∞ < a < b < ∞ and f : [a, b] → R is acontinuous function such that f (a) ≤ f (b) . Show for any y ∈ [f (a) , f (b)] , there exists a c ∈ [a, b] such thatf (c) = y.1 Hint: Let S := {t ∈ [a, b] : f (t) ≤ y} and let c := sup (S) .

1 The same result holds for y ∈ [f (b) , f (a)] if f (b) ≤ f (a) – just replace f by −f in this case.

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HomeWork #7 Due Thursday November 15, 2012

Exercise 7.1. Let f : [a, b]→ [c, d] be a strictly increasing (i.e. f (x1) < f (x2) whenever x1 < x2) continuousfunction such that f (a) = c and f (b) = d. Then f is bijective and the inverse function, g := f−1 : [c, d] →[a, b] , is strictly increasing and is continuous.

Exercise 7.2. Let ‖·‖1 , ‖·‖u , and ‖·‖2 be the three norms on Cn given above. Show for all z ∈ Cn that

‖z‖u ≤ ‖z‖1 ≤ n ‖z‖u ,‖z‖1 ≤

√n ‖z‖2 , (Hint: use Cauchy Schwarz.)

‖z‖2 ≤√‖z‖u · ‖z‖1 ≤ ‖z‖1 and

‖z‖2 ≤√n ‖z‖u .

It follows from these inequalities that ‖·‖1 , ‖·‖u , and ‖·‖2 are equivalent norms on Cn.

Exercise 7.3. Let Z denote the continuous functions on [0, 1] with values in R and as above let

‖f‖1 :=

∫ 1

0

|f (t)| dt, ‖f‖2 :=

√∫ 1

0

|f (t)|2 dt, and ‖f‖u = sup0≤t≤1

|f (t)| .

Show for all f ∈ Z that;‖f‖1 ≤ ‖f‖2 and ‖f‖2 ≤ ‖f‖u .

[Hint: for the first inequality use Cauchy Schwarz.] Also show there is no constant C <∞ such that

‖f‖u ≤ C ‖f‖2 for all f ∈ Z. (7.1)

[Hint: consider the sequence, fn (t) = tn.]

Exercise 7.4 (Continuity of integration). Let Z = C ([0, 1] ,R) be the continuous functions from [0, 1]to R and ‖·‖u be the uniform norm, ‖f‖u := sup0≤t≤1 |f (t)| . Define K : Z → Z by

K (f) (x) :=

∫ x

0

f (t) dt for all x ∈ [0, 1] .

Show that K is a Lipschitz function. In more detail, show

‖K (f)−K (g)‖u ≤ ‖f − g‖u for all f, g ∈ Z.

In this problem please take for granted the standard properties of the integral including

1. The function x → K (f) (x) is indeed continuous (in fact differentiable by the fundamental theorem ofcalculus).

2. K : Z → Z is a linear transformation.

20 7 HomeWork #7 Due Thursday November 15, 2012

3. If f (t) ≤ g (t) for all t ∈ [0, 1] , then∫ x0f (t) dt ≤

∫ x0g (t) dt for all x ∈ X.

4. From 3. it follows that∣∣∫ x

0f (t) dt

∣∣ ≤ ∫ x0|f (t)| dt.

Exercise 7.5 (Discontinuity of differentiation). Let Z be the polynomial functions in C ([0, 1] ,R) , i.e.functions of the form p (t) =

∑nk=0 akt

k with ak ∈ R. As above we let ‖p‖u := sup0≤t≤1 |p (t)| . Define

D : Z → Z by D (p) = p′, i.e. if p (t) =∑nk=0 akt

k then

D (p) (t) =

n∑k=1

kaktk−1.

1. Show D is discontinuous at 0 – where 0 represents the zero polynomial.2. Show D is discontinuous at all points p ∈ Z.

Exercise 7.6. Let fn : [0, 1] → R be defined by fn (x) = xn for x ∈ [0, 1] . Show f (x) := limn→∞ fn (x)exists for all x ∈ [0, 1] and find f explicitly. Show that fn does not converge to f uniformly.

8

HomeWork #8 Due Thursday November 29, 2012

8.1 Density Problems

Definition 8.1. Let (X, d) be a metric space. We say A ⊂ X is dense in X if for all x ∈ X, there exists{xn}∞n=1 ⊂ A such that x = limn→∞ xn. [In words, all points in X are limit points of sequences in A.] Ametric space is said to be separable if it contains a countable dense subset, A.

Exercise 8.1. Suppose that (X, d) is a separable metric space and Y is a non-empty subset of X which isalso a metric space by restricting d to Y. Show (Y, d) is separable. [Hint: suppose that A ⊂ X be a countabledense subset of X. For each a ∈ A choose {yn (a)}∞n=1 ⊂ Y so that dY (a) ≤ d (a, yn (a)) ≤ dY (a) + 1

n . Nowshow AY := ∪a∈A {yn (a) : n ∈ N} is a countable dense subset of Y.]

Exercise 8.2. Let n ∈ N. Show any non-empty subset Y ⊂ Cn equipped with the metric,

d (x, y) = ‖y − x‖ for all x, y ∈ Y

is separable, where ‖·‖ is either ‖·‖u , ‖·‖1 , or ‖·‖2 .

8.2 Series Problems

Theorem 8.2 (Integral test). Let f : (0,∞) → R be a C1 function such that f ′ (x) ≥ 0 and f ′ (x)is decreasing in x (i.e. f ′′ (x) ≤ 0 if it exists) and f (∞) := limx→∞ f (x) , see Figure 7.1. (As f is anincreasing function, f (∞) exists in (−∞,∞].) Then

[f (N + 1)− f (M)] ≤N∑

n=M

f ′ (n) ≤ [f (N + 1)− f (M)] + f ′ (M)− f ′ (N + 1) (8.1)

≤ [f (N + 1)− f (M)] + f ′ (M)

for all M ≤ N. Letting N →∞ in these inequalities also gives,

f (∞)− f (M) ≤∞∑

n=M

f ′ (n) ≤ f (∞)− f (M) + f ′ (M)

and in particular,∑∞n=1 f

′ (n) <∞ iff f (∞) <∞.

Exercise 8.3. Take f (x) = lnx in Theorem 8.2 in order to directly conclude that∑∞n=1

1n =∞.

Exercise 8.4 (This problems replaces the alternating series problem). Let 0 ≤ p <∞. Prove,

∞∑n=2

1

n (lnn)p =

{∞ if 0 ≤ p ≤ 1<∞ if p > 1

.

by applying Theorem 8.2 with the following functions;

22 8 HomeWork #8 Due Thursday November 29, 2012

1. for 0 ≤ p < 1 take f (x) = (lnx)1−p

,2. for p = 1 take f (x) = ln lnx, and

3. for p > 1 take f (x) = − (lnx)1−p

.

Exercise 8.5. Let (X, d) be a metric space. Suppose that {xn}∞n=1 ⊂ X is a sequence and set εn :=d(xn, xn+1). Show that for m > n that

d(xn, xm) ≤m−1∑k=n

εk ≤∞∑k=n

εk.

Conclude from this that if∞∑k=1

εk =

∞∑n=1

d(xn, xn+1) <∞

then {xn}∞n=1 is Cauchy. Moreover, show that if {xn}∞n=1 is a convergent sequence and x = limn→∞ xn then

d(x, xn) ≤∞∑k=n

εk.

Exercise 8.6. Prove Theorem 7.9. Namely if (X, ‖ · ‖) is a Banach space and {xk}∞k=1 ⊂ X is a sequence,

then∞∑k=1

‖xk‖ <∞ implies∞∑k=1

xk is convergent.

Exercise 8.7. For every p ∈ N, show∑∞n=0 (n)

n/pzn is convergent iff z = 0.

Exercise 8.8. Show that each of the following power series have an infinite radius of convergence and hencedefine continuous functions from C to C.

1. ez :=∑∞n=0

zn

n! ,

2. sin (z) :=∑∞n=0 (−1)

2n+1 z2n+1

(2n+1)! , and

3. cos (z) :=∑∞n=0 (−1)

2n z2n

(2n)! .

Exercise 8.9 (Inverting perturbations of the identity). For ‖A‖2 < 1 in CJN×JN , I − A is invertibleand

(I −A)−1

=

∞∑n=0

An

where the sum is absolutely convergent. Moreover the function A → (I −A)−1

is continuous on the ball,B := {A : ‖A‖2 < 1} .

8.3 Additional Practice Exrcises (Not to be collected)

Exercise 8.10 (Not to be collected). For x, y ∈ R, let

d (x, y) :=

{1 if x = y0 if x 6= y

.

Exercise 8.11 (Not be collected). Let ‖·‖ be a norm on Rn such that ‖a‖ ≤ ‖b‖ whenever 0 ≤ ai ≤ bifor 1 ≤ i ≤ n.1 Further suppose that (Xi, di) for i = 1, . . . , n is a finite collection of metric spaces and forx = (x1, x2, . . . , xn) and y = (y1, . . . , yn) in X :=

∏ni=1Xi, let

d(x, y) = ‖(d1 (x1, y1) , d2 (x2, y2) , . . . , dn (xn, yn))‖ .

Show (X, d) is a metric space.

1 For example we could take ‖·‖ to be ‖·‖u , ‖·‖1 , or ‖·‖2 on Rn. Not all norms satisfy this assumption though. Forexample, take ‖(x, y)‖ = |x− y|+ |y| on R2, then ‖(x, y)‖ is decreasing in x when 0 ≤ x < y.

8.3 Additional Practice Exrcises (Not to be collected) 23

Exercise 8.12 (Not to be collected). Suppose (X, ρ) and (Y, d) are metric spaces and A is a dense subsetof X.

1. Show that if F : X → Y and G : X → Y are two continuous functions such that F = G on A thenF = G on X.

2. Now suppose that (Y, d) is complete and f : A→ Y is a function which is uniformly continuous (Notation6.43). Recall this means for every ε > 0 there exists a δ > 0 such that

d (f (a) , f (b)) ≤ ε for all a, b ∈ A with ρ (a, b) ≤ δ. (8.2)

Show there is a unique continuous function F : X → Y such that F = f on A.Hint: Define F (x) = limn→∞ f (xn) where {xn}∞n=1 ⊂ A is chosen to converge x ∈ X. You must showthe limit exists and is independent of the choice of sequence {xn}∞n=1 ⊂ A which converges for x.

3. Let X = R = Y and A = Q ⊂ X, find a function f : Q→ R which is continuous on Q but does notextend to a continuous function on R.

[This exercise is a good test of many of the major concepts of convergence and continuity. I highlyrecommend you give it a go.]