MATH 361
Shuyi Weng
Real Analysis
September 11, 2013
Assignment 3
Exercise 1
The greatest lower bounds of the sets are
(a) 7;
(b) + 1;
(c) .
Exercise 2
The least upper bounds of the sets are
(a) 8;
(b) + 1.
Exercise 3
Example. A = cfw_x R | x = 1 2n or x
MATH 361
Shuyi Weng
Real Analysis
September 27, 2013
Assignment 6
Exercise 1
Proposition. The sequence of partial sums cfw_1 + 1/2 + 1/3 + + 1/n of the harmonic series
n=1
is not Cauchy.
1
Proof. Let = . Let N N. Let n = N + 1 and let m = 2n. Thus
2
1
1
1
MATH 361
Shuyi Weng
Real Analysis
September 23, 2013
Assignment 5
Exercise 1
Proposition. Let cfw_sn be a sequence of positive numbers. Let x be a real number with
n=1
0 < x < 1. If sn+1 < xsn for all n N, then lim sn = 0.
n
Proof. Suppose that sn+1 < xs
MATH 361
Shuyi Weng
Real Analysis
September 15, 2013
Assignment 4
Exercise 1
1
Proposition. cfw_n n does not have a limit.
n=1
1
Proof. We use Proof by Contradiction. Assume that cfw_sn = cfw_n n has a limit L. Thus
n=1
for all
|N
1
N
> 0, there exist
MATH 361
Shuyi Weng
Real Analysis
September 8, 2013
Assignment 2
Exercise 2.3.3
Proposition. For any a R, let a3 denote a a a.
Let x, y R.
1. If x < y, then x3 < y 3 .
2. There are c, d R such that c3 < x < d3 .
Proof. Suppose that x < y. There are three
MATH 361
Shuyi Weng
Real Analysis
October 9, 2013
Assignment 7
Exercise 1
Proposition. A conditionally convergent series has a rearrangement that diverges.
Proof. Suppose that
|an | diverges. Let
an is a conditionally convergent series. Thus
n=1
n=1
P be
MATH 361
Shuyi Weng
Real Analysis
October 22, 2013
Assignment 8
Exercise 1
Proposition. The characteristic function Q : R R is not continuous at any point in R.
Proof. We show this by contradiction.
Suppose that Q is continuous at c R. We have two cases:
MATH 361
Shuyi Weng
Real Analysis
December 3, 2013
Assignment 13
Exercise 1
Which of the following functions are Riemann-integrable on the interval [0, 1]?
(1) The characteristic function of the set cfw_0, 1/10, 2/10, . . . , 1.
(2) The function dened by
MATH 361
Shuyi Weng
Real Analysis
November 20, 2013
Assignment 12
Exercise 1
Proposition. Let f be a continuous function on [0, 1]. Let n = cfw_0, 1/n, . . . , n/n, and let x be
k
any point on the interval [(k 1)/n, k/n]. Then
L[f, n ]
1
n
n
1
n n
n
f (x
MATH 361
Shuyi Weng
Real Analysis
November 13, 2013
Assignment 11
Exercise 1
Proposition. Let f : [a, b] R be dierentiable and suppose that f (x) > 0 for all x [a, b]. If is the
inverse function for f , then is continuous on the interval [f (a), f (b)].
P
MATH 361
Shuyi Weng
Real Analysis
November 2, 2013
Assignment 9
Exercise 1
Proposition. If f : [a, b] R is a function and the absolute value of the slope of every
secant line on the graph of f is less than 1, then f is uniformly continuous on [a, b].
Proo
MATH 361
Shuyi Weng
Real Analysis
November 8, 2013
Assignment 10
Exercise 1
Proposition. Let f : R R be given by
x
f (x) =
sin x
if x Q
if x Q
/
Then f (0) = 1.
Proof. In order to show that f (0) = 1, we need to prove that the limit
f (x) f (0)
= 1.
x0
x0
MATH 361
Shuyi Weng
Real Analysis
September 4, 2013
Assignment 1
Part 1
Proof. Suppose that A = B. Then R A = R B. It is obvious to see that the domains
and the codomains of A and B agree. Let x A, then x B. Thus
1 if x A,
1 if x B,
A (x) =
=
= B (x)
0 if