MATH 140A FALL 2015 MIDTERM 2
Instructions: Justify all of your answers, and show your work. You may quote basic
theorems proved in the textbook or in class, unless the point of the problem is to reproduce
the proof of such a theorem, or the problem tells
M ATH 140 A - HW 7 S OLUTIONS
Problem 1 (WR Ch 4 #14). Let I = [0, 1] be the closed unit interval. Suppose f is a continuous
mapping of I into I . Prove that f (x) = x for at least one x I .
Solution. Let g (x) = x f (x), which is also continuous. If g (x
Math 140a - HW 1 Solutions
Problem 1 (WR Ch 1 #1). If r is rational (r = 0) and x is irrational, prove that r + x and
rx are irrational.
Solution. Given that r is rational, we can write r =
a
b
for some integers a and b. We are
also given that x is irrati
M ATH 140 A - HW 3 S OLUTIONS
Problem 1 (WR Ch 2 #12). Let K R1 consist of 0 and the numbers 1/n, for n = 1, 2, 3, . . . Prove
that K is compact directly from the denition (without using the Heine-Borel theorem).
Solution. Let cfw_G be any open cover of
M ATH 140 A - HW 6 S OLUTIONS
Problem 1 (WR Ch 3 #21). Prove the following analogue of Theorem 3.10(b): If cfw_E n is a sequence
of closed nonempty and bounded sets in a complete metric space X , if E n E n+1 , and if
lim diam E n = 0,
n
then
1 En
Soluti
M ATH 140 A - HW 4 S OLUTIONS
Problem 1 (WR Ch 3 #1). Prove that convergence of cfw_s n implies convergence of cfw_|s n . Is the converse true? (Assume we are working in Rk )
Solution. Let a, b Rk . Then by the triangle inequality,
|a| = |a b + b| |a b|
M ATH 140 A - HW 5 S OLUTIONS
Problem 1 (WR Ch 3 #8). If
a n converges, and if cfw_b n is monotonic and bounded, prove that
a n b n converges.
Solution. Theorem 3.42 states that if
(a) the partial sums of
a n form a bounded sequence;
(b) b 0 b 1 b 2 ;
(c
MATH 140A Fall Semester 2003 Exam I September 29, 2003 ANSWERS: 1. A; 2. B; 3. C; 4. D; 5. A; 6. E; 7. D; 8. C; 9. C; 10. E. 11. a. T; b. F; c. T; d. T. 12. a. T; b. T; c. T; d. F; e. T. 13. Removable discontinuity at x = 1; Innite discontinuity at x = 1;
MATH 140A
MIDTERM EXAMINATION III
November 18, 2003
Name
ID #
Section #
There are 10 multiple choice questions, 5 True/False questions, and 3 free response questions. To receive full credit for free response questions (problems 12, 13, and 14) all work mu
Math 140a - HW 2 Solutions
Problem 1 (WR Ch 1 #2). A complex number z is said to be algebraic if there are integers
a0 , . . . , an not all zero, such that
a0 z n + a1 z n1 + + an1 z + an = 0.
()
Prove that the set of all algebraic numbers is countable.
S
Math 140A - Fall 2014 - Final Exam
Problem 1.
Let cfw_an n1 be an increasing sequence of real numbers.
(i) If cfw_an has a bounded subsequence, show that cfw_an is itself bounded.
(ii) If cfw_an has a convergent subsequence, show that cfw_an is itself
MATH 140A FALL 2015 MIDTERM 1 SAMPLE SOLUTIONS
1. (a) (5 pts). Carefully dene the following:
(i). What it means for a set X with a distance function d to be a metric space.
(ii). What it means for p X to be a limit point of a subset E of X.
(iii). The clo
MATH 140A FALL 2015 MIDTERM 1
1. (a) (5 pts). Carefully dene the following:
(i). What it means for a set X with a distance function d to be a metric space.
(ii). What it means for p X to be a limit point of a subset E of X.
(iii). The closure E of a subse
MATH 140A FALL 2015 MIDTERM 2 SOLUTIONS
1 (10 pts). Decide if the following series converges or not. Justify your answer.
n=0
n
n2 + 5
n
n
1
Solution. Since n + 5 > n for all n, we have 2
< 2 = 3/2 for all n 0. Since
n +5
n n
1
n
also converges by the
is
Name:
PID:
Math 140A: Final Exam
Foundations of Real Analysis
You have 3 hours.
No books and notes are allowed.
You may quote any result stated in the textbook or in class.
You may not use homework problems (without proof) in your solutions.
1. (10 po
Name:
PID:
Math 140A: Midterm 2
Foundations of Real Analysis
You have 50 minutes.
No books and notes are allowed.
You may quote any result stated in the textbook or in class.
You may not use homework problems (without proof) in your solutions.
1. (10
Name:
PID:
Math 140A: Midterm 1
Foundations of Real Analysis
You have 50 minutes.
No books and notes are allowed.
You may quote any result stated in the textbook or in class.
You may not use homework problems (without proof) in your solutions.
1. (10
Math 140A - Fall 2014 - Midterm I
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation will
Math 140A - Fall 2014 - Midterm II
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation wil
MATH 140A Fall 2002 Exam III November 18, 2002 ANSWERS: 1. (D); 2. (B); 3. (B); 4. (D); 5. (A); 6. (C); 7. (D); 8. (B); 9. (E); 10. (A); 11. (D). Note: Handdrawn graphs for problem 5 are unavailable in the exam pdf le. 12. (F); 13. (T); 14. (F); 15. (F);
MATH 140A Fall 2003 Exam III November 18, 2003 ANSWERS: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. a) F [none] The correct answer is 6x2 sin(x3 + 3 ) cos(x3 + 3 ). A B A D A D C B A b) F c) T d) T e) T
12. a) f (x) is increasing on the intervals (, 0) and (8, );
MATH 140A
MIDTERM EXAMINATION II
October 27, 2003
Name
ID #
Section #
There are 8 multiple choice questions, 10 True/False questions, and 4 free response questions. To receive full credit for free response questions (problems 10, 11, 12 and 13) all work m
Math 140, Winter
Instructor: Professor Ni
Practice problems
February, 2010
1. Prove that if x is a real number and |x| < 1, then limn xn = 0.
12
2. The Cantor set are constructed as following. Let K0 = [0, 1], remove ( 3 , 3 ) let K1 =
1
2
1
2
[0, 3 ] [ 3
Math 140, Winter Instructor: Professor Ni 1. Exercise 8 of Ch4. 2. Exercise 14 of Ch4. 3. Exercise 18 of Ch4. 4. Exercise 19 of Ch4. 5. Find the limit
Practice problems
February, 2010
1+x 1x lim . x0 (1 + x)1/3 (1 x)1/3
6. Assume that the sequence cfw_an
Math 140A Test 2
100 points
November 23, 2009
Professor Evans
Directions: Show all work. In your proofs, state where you use the hypotheses. Notation: Throughout, z and an (n = 0, 1, 2, 3, . . . ) are complex. (If you can only handle the real case, you st
Assignments for Math 140A, Fall 2011
Problem Set 1. Due Friday, September 30
Rudin, Chapter 1, pp 21-22: #1, #4, #5
Also do the following problems:
A1. If A, B , and C are subsets of X , prove (carefully) that
(A \ B ) (C \ B ) = (A C ) \ B )
Notation: (A
Assignments for Math 140A, Fall 2011
Problem Set 1. Due Friday, September 30
Rudin, Chapter 1, pp 21-22: #1, #4, #5
Also do the following problems:
A1. If A, B , and C are subsets of X , prove (carefully) that
(A \ B ) (C \ B ) = (A C ) \ B )
Notation: (A