TEST 2 FOR 18.100B AND 18.100C,
NOVEMBER 12, 2009, 7:30-8:30 PM
PRINTED Name:
Write your name on EVERY page.
Do all your work on these pages.
No books or notes may be used.
Each problem is worth 20 points.
(T2.1) Let f : X Y be a bijection between com
TEST 1 FOR 18.100B AND 18.100C, FALL 2009
SOLUTIONS
Write your name on EVERY page. Do all your work on these pages. No
books or notes may be used.
(1) Let R be the set of real numbers with the standard metric. Suppose E R
has the property that for every
HOMEWORK 8 FOR 18.100B AND 18.100C, FALL 2009
DUE FRIDAY, NOVEMBER 6 AT NOON IN 2-108.
HW8.1 (Rudin Chap 4, Prob 1) Let f : R R be a function such that |f (x)
f (y )| (x y )2 for every x, y R. Prove that f is constant. Do NOT use
derivatives.
Solution: S
HOMEWORK 7 FOR 18.100B AND 18.100C, FALL 2009
SOLUTIONS.
In the rst three questions, f : X Y is a continuous map between metric
spaces.
(1) Rudin Chap 4, No 2. Show that if E X and f (E ) Y is its image under
f then the closures satisfy
f (E ) f (E ).
Giv
HOMEWORK 6 FOR 18.100B AND 18.100C, FALL 2009
DUE FRIDAY, OCTOBER 23 IN 2-108.
HW6.1 Rudin, Chap. 3, Problem 16: Fix a positive number . Choose x1 >
and dene x2 , x3 , . . . by the recursion formula
xn+1 =
1
2
xn +
xn
.
a) Prove that cfw_xn decreases mono
HOMEWORK 5 FOR 18.100B AND 18.100C, FALL 2009
SOLUTIONS.
HW5.1 Rudin Chap 3, Prob 1: Prove that convergence of cfw_sn , sn C implies
convergence of cfw_|sn |. Is the converse true? (Justify your answer).
Solution: Let s = lim sn . We check that cfw_|sn |
HOMEWORK 4 FOR 18.100B AND 18.100C, FALL 2009
SOLUTIONS (DECIEDLY PEDANTIC).
As usual the problems will each be worth 10 points and clarity is especially
prized.
HW4.1 Rudin Chap 2, 22:- A metric space is said to be separable if it contains a
countable de
HOMEWORK 3 FOR 18.100B AND 18.100C, FALL 2009
DUE FRIDAY, SEPTEMBER 25 IN 2-108.
HW3.1 Rudin, Chap. 2, Problem 6: Let E be the set of all limit points of a set E .
Prove that E is closed. Prove that E and E have the same limit points.
Do E and E have the
HOMEWORK 2 FOR 18.100B AND 18.100C, FALL 2009
DUE FRIDAY, SEPTEMBER 18 AT NOON IN 2-108.
HW2.1 Rudin Chap 1, Prob 13: If x and y are complex numbers show that
|x| |y | |x y |.
Solution: We can assume that |x| |y | otherwise we can interchange
the roles of
HOMEWORK 1 FOR 18.100B AND 18.100C, FALL 2009
SOLUTIONS
This rst assignment is due on Friday September 11, after only one lecture. It
is very important that your solutions to these problems be written out clearly. For
that reason we are giving you hints (