MATH 3033 Real Analysis

Homework 1

Due on Feb 26, 2021

1. Let S and T be sets in Rn.

(a) Prove that if S ⇢ T , then Int(S) ⇢ Int(T ), S ⇢ T , and S0 ⇢ T 0.

(b) Prove that Int(S) is the largest open set in Rn contained in S, i.e. if U is any open set in Rn

contained in S, then U ⇢ Int(S).

(c) Prove that a 2 S0 if and only if there is a sequence {xn} in S such that xn 6= a for any n and

lim

n!1xn = a.

2. Let A be a set in Rn. Prove each of the following:

(a) Rn \ Int(A) = Rn \A

(b) Int(A) = A \ @A

(c) A is closed if and only if @A ⇢ A

(d) A is open if and only if A \ @A = ;.

3. Give an example of open covering of the open interval (0, 1) in R which has no finite subcovering.

4. Prove each of the following:

(a) If A is a non-empty collection of compact sets in Rn, then

\

A2A

A is compact.

(b) If A1, . . . , An are compact subsets in Rn, then

n[

i=1

Ai is compact.

5. Let S be an non-empty set in Rn. Prove that S is compact if and only if every sequence {xm} has a

subsequence {xmk} which converges in S i.e. there exists x 2 S such that lim

k!1

xmk = x.

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MATH 3033 Real Analysis

Homework 2

Due on March 19, 2021

1. Let U ⇢ Rn be open and f : U ! Rm be a continuous function on U . Prove or disprove each of the

following statements:

(a) If O ⇢ U is an open set in Rn, then f(O) is open in Rm.

(b) If B ⇢ U is a bounded set in Rn, then f(B) is bounded in Rm.

(c) If {xk} is a Cauchy sequence in U , then {f(xk)} is a Cauchy sequence in Rm.

2. Let f : R2 ! R be defined by f(x, y) =

8<:

xy2

x2 + y4

if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0)

(a) Prove that the directional derviative of f along any direction exists.

(b) Prove that lim

(x,y)!(0,0)

f(x, y) does not exist and hence f is not continuous at (0, 0).

3. Let K ⇢ Rn be compact, V ⇢ Rm, and f : K ! V be a bijective function such that it is continuous

on K. Prove that f1 : V ! K is continuous on V .

4. Let f : R2 ! R be defined by

f(x, y) =

8><>:(x

2 + y2) sin

1p

x2 + y2

!

if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0)

Show that

(a)

@f

@x

,

@f

@y

are discontinuous at (0, 0).

(b) f is di↵erentiable everywhere.

5. Let f : R3 ! R2 and g : R2 ! R3 be defined by

f(x, y, z) = (x2 + 3y z, 2x y + z2)

g(u, v) = (u2 + v, 3v, u+ v2)

(a) Find Df(x, y, z) and Dg(u, v).

(b) Compute D(f g)(1, 1).

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MATH 3033 Real Analysis

Homework 3

Due on April 9, 2021

1. Consider f(x1, x2, y1, y2) = (w1, w2) where

w1 = x

2

1 5x22 + 4x21y1y2

w2 = 2x1y

2

2 3y21y22 + x2

(a) Using Implicit Function Theorem, prove that f(1, 1, 1, 1) = (0, 0) and there exists open sets

U and V in R2 containing (1, 1), a continuously di↵erentiable function g : U ! V such that

f(x1, x2, g(x1, x2)) = (0, 0) for all (x1, x2) 2 U .

(b) Write down explicitly the Jacobian matrix Dg(x1, x2)

2. Let {fn} be a sequence of monotone functions on [a, b]. Prove that

1X

n=1

fn converges uniformly on [a, b]

if both the series

1X

n=1

fn(a) and

1X

n=1

fn(b) converge absolutely.

3. Let fn(x) =

1

x

+

1

n

, x 2 (0, 1).

(a) Prove that {fn} converges uniformly on (0, 1).

(b) Is it true that {fngn} converges uniformly on D if both {fn} and {gn} converge uniformly on D?

(Hint: Consider f2n(x) where fn(x) is defined in part (a).)

4. Discuss the convergence (pointwise/uniform) of the following sequences of functions on (0, 1):

(a) fn(x) =

1

nx+ 1

, n = 1, 2, . . .;

(b) fn(x) =

x

nx+ 1

, n = 1, 2, . . .;

(c) fn(x) = ncx(1 x2)n, n = 1, 2, . . ., c 2 R.

(Hint: For each fixed n, find xn 2 (0, 1) such that fn(xn) is maximum and consider lim

n!1 fn(xn).)

5. (a) Find the radius of convergence of

1X

n=1

x2n1

2n 1 .

(b) Let S(x) =

1X

n=1

x2n1

2n 1 , for |x| < ⇢, where ⇢ is the radius of convergence found in part (a).

Evaluate S0(x) and hence prove that S(x) =

1

2

ln

✓

1 + x

1 x

◆

, for x 2 (⇢, ⇢)

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MATH 3033 Real Analysis

Homework 4

Due on April 25, 2021

1. Let f(x) = ex

2

.

(a) Express f(x) as a power series centered at 0.

(b) Prove that

Z 1

0

f(x) dx =

1X

k=0

(1)k

(2k + 1)k!

.

(c) Let sn =

nX

k=0

(1)k

(2k + 1)k!

and En(x) = sn

Z 1

0

f(x) dx. Prove that

|En(x)| 1

(2n+ 3)(n+ 1)!

.

2. Consider the power series

1X

n=0

(1)n1 (2n)!

22n(n!)2(2n 1)x

n.

(a) Prove that the radius of convergence is 1 and

p

1 + x =

1X

n=0

(1)n1 (2n)!

22n(n!)2(2n 1)x

n on |x| < 1.

(b) Show that

p

2 =

1X

n=0

(1)n1 (2n)!

22n(n!)2(2n 1) .

3. Let A1 = [0, 1] \

1

3 ,

2

3

be the subset of [0, 1] obtained by removing those points which lie in the open

middle third of [0, 1]; i.e. A1 =

⇥

0, 13

⇤ [ ⇥ 23 , 1⇤. Let A2 be the subset of A1 obtained by removing

the open middle third of

⇥

0, 13

⇤

and of

⇥

2

3 , 1

⇤

. Continue this process and define A3, A4, . . .. The set

C =

1\

n=1

An is called the Cantor set.

(a) C is a compact set.

(b) C has measure zero.

(c) C =

( 1X

n=1

an

3n

an 2 {0, 2} for all n 2 N

)

.

4. Let µ⇤ be the (Lebesgue) outer measure.

(a) If A,B ⇢ R and µ⇤(A) = 0, then prove that µ⇤(A [ B) = µ⇤(B). (Note: B may not be a

measurable set.)

(b) Prove that if E ⇢ R is measurable and µ⇤(E) < 1, then for any ✏ > 0, there exists an open set

O ⇢ R such that E ⇢ O and µ⇤(O \ E) < ✏.

5. Suppose {fn} is a sequence of measurable functions on R. Prove that the set of points x at which

{fn(x)} converges is measurable.

(Hint: For m,n, k 2 N, consider the set x 2 R | |fn(x) fm(x)| < 1k .)

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MATH 3033 Real Analysis

Homework 5

Due on May 9, 2021

1. Find an example in which strict inequality of Fatou’s lemma occurs.

2. Let E 2 M and f be a nonnegative measurable function that is integrable on E. Prove that for any

✏ > 0, there exists > 0 such that for every meaurable set A ⇢ E with µ(A) < , we haveZ

A

f dµ < ✏.

(Hint: Define a sequence of functions {fn} where fn(x) =

(

f(x) when f(x) n

n when f(x) > n

).

3. Compute the following limits if they exist and justify the calculations:

(a) lim

n!1

Z 1

0

⇣

1 +

x

n

⌘n

sin

⇣x

n

⌘

dx

(b) lim

n!1

Z 1

0

n2xen

2x2

1 + x

dx

(c) lim

n!1

Z 1

1

(sinx)n

x2

dx

4. Suppose E 2 M. Let {gn} be a sequence of Lebesgue integrable functions which converges a.e. to a

Lebesgue integrable function g. Let {fn} a sequence of measurable functions which converges a.e. to

a measurable function f . Suppose further that |fn| gn a.e. on E for all n 2 N. Prove thatZ

E

g dµ = lim

n!1

Z

E

gn dµ)

Z

E

f dµ = lim

n!1

Z

E

fn dµ.

(Hint: Mimic the proof of LDCT.)

5. Consider the function f : [0,1)! R defined by

f(x) =

8<:

sinx

x

when x > 0

1 when x = 0.

(a) Prove that the improper Riemann integral

Z 1

0

f(x) dx converges.

(Hint: Use integration by parts.)

(b) Prove that f is NOT Lebesgue integrable on [0,1).

(Hint: Consider lim

n!1

Z

[1,n]

| sinx|

x

dµ(x).)

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