Math H104
Homework 1 Solutions
9/1/2016
Exercises:
1. Read Chapter I Notions from Set Theory in Rosenlicht.
2. Let f : X Y be a function.
(a) For a subset A X, show f 1 (f (A) A.
(b) Show that f is one-to-one iff f 1 (f (A) = A for all A X.
(c) For a subs
Math H104
Homework 3 Solutions
9/15/2016
Exercises:
1. Consider the map d1 : R2 R2 R defined for ~x = (x1 , x2 ) and ~y = (y1 , y2 ) in R2 as
d1 (~x, ~y ) = |x1 y1 | + |x2 y2 |.
(a) Prove that (R2 , d1 ) is a metric space.
(b) In the Euclidean plane R2 an
Math H104
Homework 2 Solutions
9/8/2016
Exercises:
1. Prove that if S R is non-empty and bounded below, then it has an infimum.
2. For S R a non-empty subset that is bounded above and x R, let xS be the set cfw_xs : s S.
(a) Show that if x > 0, then sup(x
MATH H104 LECTURE 23, NOVEMBER 15, 2005
LECTURER: YUVAL PERES.
SCRIBE: BRIAN SHOTWELL
We move on to your 2nd (or 3rd) exposure to Calculus.
Denition 0.1. The derivative of a function f (x) is denoted by f (x), which is dened by
f (x + h) f (x)
= L.
h0
h
(
MATH H104 LECTURE, NOVEMBER 2, 2005
LECTURER: YUVAL PERES.
SCRIBE: ORIE SHELEF
Theorem 0.1. Let X,Y be metric spaces. If fn : X Y are continuous and fn converges
to f : X Y uniformly then f is continuous.
Denition 0.2. (Reminder - Uniform Convergent) fn f
MATH H104 LECTURE 20, NOVEMBER 10, 2005
LECTURER: YUVAL PERES. SCRIBE: BERNARD LIANG
Denition 20.1: A metric space X is separable if there is a countable set D X which is dense in X. (Examples: R is separable. Rn is separable; Qn Rn is dense.) More genera
MATH H104 LECTURE 24, NOVEMBER 17, 2005
LECTURER: YUVAL PERES.
SCRIBE: DAVID WONG
Theorem 0.1 (Darbouxs Theorem). Given f dierentiable on [a, b], dened in a neighborhood of [a, b], f attains any value between f (a) and f (b).
Proof. Suppose f (a) < f (b).
MATH H104 LECTURE 1, AUGUST 30, 2005
LECTURER: YUVAL PERES.
SCRIBE: THOMSON NGUYEN
Let Z be the set of all integers cfw_0, 1, 1, 2, 2, . . . and let N be the positive integers
cfw_1, 2, 3, . . .. Denote the rational numbers by Q = cfw_ a : a, b Z and b =
Math H104: Honors Introduction to Analysis
Fall 2005
Lecture 1: August 30
Lecturer: Yuval Peres
Scribe: Thomson Nguyen
Let Z be the set of all integers cfw_0, 1, 1, 2, 2, . . . and let N be the positive integers
cfw_1, 2, 3, . . .. Denote the rational num
MATH H104 LECTURE 25, NOVEMBER 22, 2005
LECTURER: YUVAL PERES.
SCRIBE: JONATHAN GOLDMAN
Convergence of Series. Suppose > 0. Then
(1)
n=1
1 < for > 1
n = for 1
Blocking Test. Suppose cfw_an n1 is a sequence which is decreasing (weakly) and an > 0, n.
Then
MATH H104 LECTURE XX, NOVEMBER 29, 2005
LECTURER: YUVAL PERES.
SCRIBE: KE LU
Read compelet proof of taylor approximation in text. Recap, points reached:
Rn (h) = f (x + h) Pn (h)
n
f (k) (x)
Pn (h) =
hk
k=0
k!
If f C n+1 (a, b) and x (a, b) then there ex
The Completion of a Metric Space
Brent Nelson
Let (E, d) be a metric space, which we will reference throughout. The purpose of these notes is to guide
you through the construction of the completion of (E, d). That is, we will construct a new metric space,
Math H104
Syllabus
Instructor
Brent Nelson
Evans 851
brent [at] math.berkeley.edu
Fall 2016
Lecture
Tuesdays and Thursdays
11:00 am - 12:30 pm
Cory 289
Office Hours: Tuesdays 3:30 pm - 5:30 pm, Wednesdays 11:00 am - 12:30 pm, and by
appointment.
Course We
Solutions
Math H104
Midterm 1
1. (a) (2 pts) The sequence (xn )nN converges to x in (E, d) if > 0, N N so that n N , d(xn , x) < .
p
(b) (8 pts) We claim the limit is 2. Let > 0. Choose N > 4/. Then for any n N we have n42 N42 < and
thus
3
2n3 + 6
2n
Math H104
Midterm 1
Name:
Student ID Number:
Instructions:
1. You will have 80 minutes to complete the exam.
2. The exam is a total of 5 questions and each question is worth 10 points.
3. Unless stated otherwise, you may use results we proved in class and
Homework 4 Solutions
Math H104
9/22/2016
Exercises:
1. Let A, B be subsets in a metric space (E, d).
(a) If A B, show A B and A B .
(b) Show A B = A B and (A B) = A B .
(c) In E = R with the usual metric, give examples showing A B 6= A B and (A B) 6= A B