Ma 4111: Advanced Calculus
Solutions to Homework Assignment 6
Prof. Wickerhauser
Due Tuesday, November 20th, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write
Answers to Homework 5, Math 4111
Do not submit problems in blue, but at least do them.
(1) Prove that the only subsets of Rn which are both open and
closed are Rn and the empty set. (Hint: If X is another such a
set, pick a X, b X and consider supcfw_t [0
Answers to Homework 4, Math 4111
(1) Prove that the following examples from class are indeed metric
spaces. You only need to verify the triangle inequality.
(a) Let C be the set of continuous functions from [0, 1] R
with the sup norm: |f | = supx[0,1] cfw
Answers to Homework 9, Math 4111
(1) Prove that the map f : [0, ) [0, ) given by f (x) = xn ,
n N is a homeomorphism. Denoting the inverse as g(x) =
1
x n , prove that g is dierentiable at every point in (0, ) and
calculate its derivative.
We already know
Answers to Homework 8, Math 4111
(1) Let C be the set of continuous functions on the closed interval
[0, 1] with sup norm. Dene a function ev : C R (usually
called the evaluation map) by ev(f ) = f (0) for any f C.
Prove that ev is uniformly continuous.
G
Answers to Homework 6, Math 4111
Do not submit problems in blue, but at least attempt them.
(1) Prove that any continuous map from f : R Q (with the usual
metrics) is constant.
If it is not constant, let f (a) < f (b). Since the inclusion Q
R is continuo
Answers to Homework 11, Math 4111
(1) Let f C ([a, b]) such that |f (n) (x)| M for all x [a, b] and
for all n 0. Prove that there exists polynomials Pd for d 0
such that lim Pd = f in the sup norm.
This is just an application of Taylors theorem. Consider
Homework 10, Math 4111, due 14 Nov 2013
Do not submit problems in blue, but at least attempt them.
(1) Prove that exp is dierentiable at 0 and exp (0) = 1.
We have done all the estimates necessary for this before. Let
me recall it again:
Let M > 0 be xed.
Answers to Homework 12, Math 4111
Do not submit problems in blue, but at least attempt them.
(1) Let be the function on [a, b] dened as (x) = 0 if x is rational
and (x) = 1 if x is irrational. Prove that if f R(), then
f is constant. (Hint: It may be easi
Answers to Homework 3, Math 4111
(1) If A is any set, dene P(A), the power set of A to be the set of
all subsets of A. Prove that if A is a set with cardinality n N,
then the power set P(A) has cardinality 2n .
We use induction on n. If n = 1, A = cfw_a,
Answers to Homework 7, Math 4111
(1) Let f : R R be a continuous monotonic (increasing or decreasing) function. Prove that f is a homeomorphism to its
image.(Recall that f is monotonic increasing if for any x > y,
f (x) > f (y).) You may prove this assumi
Ma 4111: Advanced Calculus
Solutions to Homework Assignment 7
Prof. Wickerhauser
Due Thursday, December 6th, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write
Ma 4111: Advanced Calculus
Solutions to Homework Assignment 5
Prof. Wickerhauser
Due Tuesday, November 6th, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write u
Ma 4111: Advanced Calculus
Solutions to Homework Assignment 4
Prof. Wickerhauser
Due Tuesday, October 23rd, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write u
Ma 4111: Advanced Calculus
Solutions to Homework Assignment 2
Prof. Wickerhauser
Due Tuesday, September 25th, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write
Ma 4111: Advanced Calculus
Solutions to Homework Assignment 3
Prof. Wickerhauser
Due Tuesday, October 9th, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write up
Ma 4111: Advanced Calculus
Solutions to Homework Assignment 1
Prof. Wickerhauser
Due Tuesday, September 11th, 2012
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems but you must write
Answers to Homework 1, Math 4111, due September 5
(1) Prove that for sets A, B, C, A (B C) = (A B) (A C).
As usual, when we want to prove two sets are equal, we show
one is contained in the other and then the latter is contained in
the rst. We will only d
Homework 2, Math 4111, due September 12
(1) Let xn =
1
1
k+1 .)
k
n
xn =
k=1
n
1
k=1 k(k+1) .
1
=
k(k + 1)
Prove that lim xn = 1. (Hint:
n
1
1
k k+1
k=1
=1
1
k(k+1)
=
1
.
n+1
Rest is easy.
(2) Let xn = n k12 . Prove that lim xn exists. (Hint: The set
k=1
Real AnalysisLecture Notes for 4111
Brian E. Blank
September 12, 2016
first president university press
ii
Chapter 1. The Real Numbers
1.1
Set TheoryBasics
The reader is expected to have already absorbed the contents of a course with a title such as Founda