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
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
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 al
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 h
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
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 i
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.
S
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 c
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 int
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
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
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,
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, th