Math 131A - Section 2 Spring 2010 Homework 4
All problems graded (out of 3 points for this short assignment). 10.1(b). bounded; (d) bounded; (f) bounded, non-increasing 10.2. Let (sn ) be a bounded non-incerasing sequence. Let S = cfw_sn |n N and u = inf
Math 131A - Section 2 Spring 2010 Homework 6
The following problems were graded for this assignment: 14.7 (2 points), 14.12 (2 points), 15.6 (2 points), 17.9(b,c) (2 points), 17.10 (b,c) (2 points), 17.13 (3 points), 17.14 (2 points). 14.2.(a). For n 2, w
Math 131A Lecture 4: Homework 4, Due 10/24 in TA session
1-2. 10.6, 10.8
3. Let us consider the sequence x1 = 1 and xn+1 = 1 +
1
.
xn
(a) Show that xn [ 3 , 2] for n 2.
2
(b) Using (a), show that |xn+1 xn | 4 |xn xn1 | for n 3.
9
(c) Deduce that cfw_xn i
Math 131A - Section 2 Spring 2010 Homework 5
For this assignment, the following problems were graded: 11.5 (3 points), 11.7 (3 points), 11.9(b) (2 points), 12.1 (2 points), 12.4 (2 points), 12.12 (3 points).
1 1 11.5.(a). Let (qn ) be an enumeration of th
Math 131A-1: Homework 1
Due: April 1, 2016
1. Send me an e-mail introducing yourself. Let me know if you like to be called something
other than your registrar listing, and anything you think I should know about your background.
2. Read Sections 1-3 in Ros
M 614%ij O?
1. (a) State the Intermediate Value Theorem. (0) Suppose that x)
is continuous on [09,0], that b 6 (me) and f (a) < f (c) < b). Prove
that there exist $1,332 6 [(1, c] with :31 7 932 such that f (311) = f(m2).
L5r*hd31[) H2 (120:) IS tcmfmwes
1. Let (Sn) be a. sequence. (2) Dene: limTHw = s 6 R. (10)
Use the denition in part (a) to prove that if 3 Z 0 for all n and
limnoo = 5, then s 2 0.
(rm/cfw_#3623 gig/W49 A<o m4 M 6242
47% W &/J':NAWQKM%?N
IMF/Lax Mm ~4J< -/Jg u
A</Lw/A</OL CMQ An<0
gu JA
1. (a) Dene: limw_,1,_ f($) = L E R. (b) Prove that if an) 2 0 for
all [B 6 (1,1) and limHL an) = L then L 2 0. 2. (a) Prove that if a, b 6 R, then
Hal - lbll S |ab|.
(b) Prove that is f(:c) is continuous at 5120, then |fl(m) dened by
| f Km) = | f (33)|
1. State and prove the Archimedean Property. 2.
1i
minf 31,. = +00 then lime.r1 = +00.
(0) Dene: liminf 3 of a sequence (3). (b) Prove that if 3. (a) Dene: (Sn) converges to s 6 R. (b) Suppose that 3" 0 for
all n and (3) converges to 3 7Q 0 Prove that in
HOMEWORK 3
MATH 131A
Chapter 2, Section 9: 9.4 9.15.
The following problems are due on Friday, October 28:
Chapter 2, Section 9: 9.6, 9.9, 9.10, 9.11, 9.12.
1
1. (3.) Dene: rf(a:) is uniformly continuous on S Q R. (b) Suppose
f (cc) is uniformly continuous on S Q R and f(:z:) Z c > 0 for all
a: E S. Prove that the function 1/) dened by (1/f)(ac) : 1/f(:1:) is
also uniformly continous on 5.
Wow m cm epo W Wt gm
1.Let x) be continuous on [a, b] and differentiable on (a, 15) such that
f(:z:) is integrable on [a, (2]. Given that
L(f,P) s fa?) _f(a) S U(f,P)
for all partitions P of [01,19], prove that
f: )dw=f(b)~f re).
dz f ) cfw_oft/(MW ~(1LL7cfw_W)<G*
QM ma (we.
Math 131A-1: Homework 2
Due: April 8, 2016
1. Read Sections 4-5 in Ross.
2. Do problem 2.3, 3.4, 3.7, and 3.8 in Ross.
3. Do problems 4.1 - 4.4 in Ross for (a), (b), (m), (r), and (w). [Please do not use the
answer formats suggested by the textbook; inste
Math 131A-1: Homework 7
Due: May 13, 2016
1. Read Sections 19-20 in Ross.
2. Do problems 17.2, 17.3 (a),(c), 17.10, 17.12, 18.4, 18.7, 18.10, 19.1(a),(c),(f),(g), 19.2(b),
19.4 in Ross.
3. The stars over Babylon. For each rational number r (0, 1], write r
Math 131A-1: Homework 6
Due: May 6, 2016
1. Read Sections 17-18 in Ross.
2. Do problems 14.2 (a),(f), 14.3(b),(e), 14.4(c), 14.5, 14.8, 14.12, 14.13 in Ross.
3. Do problems 15.1, 15.4(b) in Ross. [You can use what you know about integration from
calculus
Math 131A-1: Homework 8
Due: May 20, 2016
1. Read Sections 28, 29, and 23 in Ross.
2. Do problems 20.6, 20.13, 20.16, 20.17, 28.2, 28.5, 28.8, 28.14, and 28.16 in Ross.
3. The second midterm is Wednesday, May 18 in MS 5200, 8-8:50 a.m. It will be focused
Math 131 A, Lecture 1
Real Analysis
Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward correct proo
Math 131 A, Lecture 1
Real Analysis
Midterm 2
Instructions: You have 50 minutes to complete the exam. There are five problems, worth
a total of fifty points. Write your solutions in the space below the questions. If a question is in
multiple parts, clearl
Math 131A-1: Homework 5
Due: April 29, 2016
1. Read Sections 12, 14-15 in Ross.
2. Do the the exercises in 11.2-4 in Ross for the sequences an , bn , un , xn , and zn .
3. Do exercise 12.3(a), (b), (c), and (g) in Ross.
4. Do exercises 11.5, 11.9(b), 12.4
Math 131A-1: Homework 9
Due: May 27, 2016
1. Read Sections 31-32 in Ross.
2. Do problems 29.2, 29.4, 29.5, 29.13, 29.16, 29.18, 23.1(a),(c),(e),(g), 23.5, and 31.1 in Ross.
3. The five constants. Recall that the imaginary number i satisfies the property t
MATH 131A Analysis
Theorem Sheet
If you use a numbered result on a homework or exam question, you should cite it. You can
use the other results without citing them. There will be a copy of the numbered results
attached to all of the exams.
1. Real numbers
Math 131A: Homework 4
(due Friday, February 5th)
1. Let cfw_En nN be a collection of subsets of N. Consider the set
E = cfw_n N : n
/ En .
Show that En 6= E for any n N. Deduce that the set cfw_A : A N is uncountable.
2. Consider a, b R with a < b.
(a) I
Math 131B-2
Homework 5
Due: November 8, 2016
This assignment is due on November 15th, 2016 in the discussion section. Provide complete well-written solutions
to the following exercises.
Exercise 1. Consider the following definition (which is related, but
Math 131B-2
Homework 2
Due: October, 11th, 2016
This assignment is due on October 11th, 2016 in the discussion section. Provide complete well-written solutions to
the following exercises.
Exercise 1. Let (X, d) be a metric space and Y X. A collection cfw_
HOMEWORK 5
SHUANGLIN SHAO
Abstract. Please send me an email if you find mistakes. Thanks.
1. P151. # 19.1
Proof. (a). (b). The function is uniformly continuous on [0, ] by Theorem
19.2 because it is continuous on [0, ].
(c). This function is uniformly con
HOMEWORK 1
SHUANGLIN SHAO
Abstract. Please send me an email if you find mistakes. Thanks.
1. P 5. #1.1
Proof. We prove it by math induction. For n = 1, both sides equal to 1.
Suppose the claim is true for n 2 N. We prove that it is true for n + 1. We
cons
HOMEWORK 4
SHUANGLIN SHAO
Abstract. Please send me an email if you find mistakes. Thanks.
1. P130 . # 17.1
Proof. (a). The domain of the new functions should be the intersection of
the old domains of the two functions, f + g and f g; so it is ( 1, 4]. For
HOMEWORK 2
SHUANGLIN SHAO
Abstract. Please send me an email if you find mistakes. Thanks.
1. P38. #7.3
Proof. The sequences in (b), (d), (f ), (j),(p),(r), (t) converge; their limits
are 1, 1, 1, 72 , 2, 1 and 0, respectively.
The limit in (h) diverges be