ANSWERS TO SELECTED EXERCISES: HW1-4
Exercise 2.5.10 Suppose {an }, {bn } are sequences of nonnegative real numbers with lim bn = b = 0 and lim an = a. Prove that lim an bn = ab.
n n n
Proof: By hypothesis, both {an } and {bn } are bounded, hence s
Homework 8 Solutions
Math 131B-1
(15.7.10) (a) Observe that |4n cos(32n x)| < 4n , so by the Weierstrass M -test, f (x)
converges uniformly. Since fn (x) = 4n cos(32n x) is continuous, f is as well. (b)
j
j+1
32n
For n > m, 32m is divisible by 2, so cos
Math 131 B, Lecture 1
Real Analysis
Sample Midterm 1
Instructions: You have 50 minutes to complete the exam. There are ve problems, worth a
total of fty points. You may not use any books or notes. Partial credit will be given for progress
toward correct p
Homework 1 Solutions
Math 131B-1
(3.26) The empty set contains no points, so it is trivially true that any point in has
a neighbourhood in . Ergo is open. Similarly, has no limit points, so contains all
its limit points and is closed. The entire space M
Math 131 B, Lecture 1
Real Analysis
Sample Midterm 1
Instructions: You have 50 minutes to complete the exam. There are ve problems, worth a
total of fty points. You may not use any books or notes. Partial credit will be given for progress
toward correct p
Homework 2 Solutions
Math 131B-1
(3.29) Let (M, d) be a metric space, and
d (x, y) =
d(x, y)
.
1 + d(x, y)
d(x,x)
1+d(x,x)
0
1
We check that d is a metric.
For any x M , d (x, x) =
=
= 0.
(Positivity) For any x, y M such that x = y, d(x, y) > 0 and 1 +
Homework 2 Solutions
Math 131B-1
(4.8) Let (S, d) be a compact metric space. Let cfw_xn be a sequence in S, and T =
cfw_xn : n N be its set of values. If T is nite, cfw_xn is eventually constant, hence
converges. If T is innite, T has a limit point p i
Homework 2 Solutions
Math 131B-1
(4.21) Let f : S R be continuous, and f (p) > 0. Let = f (p) , and choose such
2
that x B(p; ) implies f (x) B(f (p); ), or equivalently that |f (p) f (x)| < . In
particular f (p) f (x) < , implying that f (x) < f (p) + f
Math 131B-1: Homework 1
Due: January 13, 2014
1. Send me an e-mail letting me know
Which section of 131A you took.
What you like to be called, especially if this is dierent from what the registrar has
listed as your rst name.
Anything else you think I
Math 131B-1: Homework 3
Due: January 24, 2014
1. Read Apostol Sections 4.2-4, 2.12-13, 3.8-12.
2. Do problems 4.8, 4.9, 2.18, 2.19, 3.16, and 3.19 in Apostol.
3. Let (M, d ) be any innite set M with the discrete metric (i.e. d (x, y) = 1 when x = y).
Sho
Math 131B-1: Homework 4
Due: February 3, 2014
1. Read Apostol Sections 4.8-9, 4.11-13, 4.15-17, 4.19-20. [Most of these are short.]
2. Do problems 4.21, 4.25, 4.28, 4.33, 4.38, 4.39 in Apostol.
3. We say that a subset S of a metric space M is dense if eve
Math 131B-1: Homework 5
Due: February 10, 2014
1. Read Apostol Sections 4.19-20, 9.1-8.
2. Do problems 4.52, 4.54, 9.2, 9.3, 9.16, 9.21, 9.22 in Apostol. [Note that you may wish to
wait until after Mondays lecture to do 9.16.]
3. Dinis Theorem Let fn : X
Math 131B-1: Homework 9
Due: March 14, 2014
1. Read Tao Sections 16.5-6, Apostol Sections 12.1-5, 12.8-10.
2. Do Tao 16.5.1, 16.5.2, 16.5.3, 16.5.4. [Note the existence of a typo in 16.5.4: instead of
inf (x), it should say 2inf (x).]
3. Do Apostol 12.1,
Math 131B-1: Homework 8
Due: March 7, 2014
1. Read Tao Sections 16.1-4.
2. Do Tao exercises 15.7.10, 16.2.2.
1
3. Do Tao exercise 16.2.3 [Hint: Its important to pick a function g for which |g| = 0 g.
Try working with g such that g(x) = x2 x on [0, 1] and
Homework 6 Solutions
Math 131B-1
1
1
(9.31) By assumption, 1 = lim supn |an | n . This implies that lim supn |ak | n = 21k ,
n
2
k n
is 2k . Similarly, for the power series
so the radius of convergence of
n=0 an z
kn
the knth coecient of the power series
Homework 6 Solutions
Math 131B-1
6
(1.27) a) 2 + 2i b) 25 +
17
i
25
c) 1 + i d) 1 + i
(1.30) a) The unit circle. b) An open disk of radius 1 about the origin. c) A closed disk
of radius 1 about the origin. d) The vertical line cfw_ 1 + bi : b R. e) The
ANSWERS TO SELECTED EXERCISES: HW1
Exercise 6.1.7
a. Suppose f is continuous on [a, b] with f (x) prove that f (x) = 0 for all x [a, b]
0 for all x [a, b]. If
b a
f = 0,
b. Show by example that the conclusion may be false if f is not continuo
ANSWERS TO SELECTED EXERCISES: HW2
Exercise 6.2.12 Let f : [0, 1] R be a continuous function. Prove that
1 n
lim
f (xn )dx = f (0).
0
1
Proof: First approach. Let In =
0
f (xn )dx. It is natural to apply the mean
value theorem for the continu
ANSWERS TO SELECTED EXERCISES: HW3
Exercise 7.1.5 Determine all values of p and q for which the following series converges: 1 k q (ln k)p
k=2
Proof: The main idea is that the behavior of this series is determined by the 1 factor kq rather than (ln
ANSWERS TO SELECTED EXERCISES: HW4
Exercise 7.3.3a Prove that if then ak bk converges. Proof: Since
ak converges and
bk converges absolutely,
ak converges, we must have lim an = 0, thus in particular
n
an is bounded, say |an | < M for all n N,
ANSWERS TO SELECTED EXERCISES: HW5-7
Exercise 8.1.5 Let fn (x) = (x/n)e-x/n , x [0, ). a. Show that lim fn (x) = 0 for all x [0, ).
n
b. Given > 0, does there exist an integer n0 N such that |fn (x)| < for all x [0, ) and all n n0 . c. Answer
ANSWERS TO SELECTED EXERCISES: HW8
Exercise 8.7.7 Suppose f (x) = R > 0. For |x - c| < R, set F (x) =
k=0 x c
ak (x - c)k has radius of convergence f (t)dt. Prove that |x - c| < R.
F (x) =
k=0
ak (x - c)k+1 , k+1
Proof: Let with 0 < < R be
Exercise 2.5.7 Let {an }, {bn } be two sequences of positive terms. Prove that liman bn liman limbn ,
when the product on the right is not of the form 0 . Proof: We distinguish 3 cases:
liman = : Then by hypothesis limbn > 0, so the right-hand si
Math 131b-Midterm
Name:
1. (25 points) Indicate whether each of the following statements is true or
false. A correct response will receive full credit. An incorrect response may
receive partial credit if accompanied by a reasonable explanation. Read
each
MATH 131B
1ST PRACTICE MIDTERM
Problem 1. State the books denition of:
(a) Uniform convergence of a sequence of function
(b) A metric
(c) Pointwise convergence of a sequence of functions
(d) A continuously differentiable function of two variables
(e) A co
MATH 131B
1ST PRACTICE MIDTERM
SOLUTIONS
Problem 1. State the books denition of:
(a) Uniform convergence of a sequence of function
(b) A metric
(c) Pointwise convergence of a sequence of functions
(d) A continuously differentiable function of two variable
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