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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