Supplements to the Exercises in Chapters 1-7 of Walter Rudins
Principles of Mathematical Analysis, Third Edition
by George M. Bergman
This packet contains both additional exercises relating to the material in Chapters 1-7 of Rudin, and
information on Rudi
Homework 5 Math 118B, Winter 2010
Due on Thursday, March 4th, 2010
1. Show that if f 0 and if f is monotonically decreasing, and if
n
n
cn =
k=1
f (k )
f (x) dx,
1
then
lim cn
n
exists.
2. Let n (x) be positive-valued and continuous for all x [1, 1], wit
Self-Assessment Questions Math 118A, Fall
2009
1. Evaluate the limit
n
lim n
n
e
1
1+
n
n
.
without using LHospitals rule.
2. Prove that
n
1
1+
n
lim
n
exists using the following steps:
(a) Prove that
n
an =
1
1+
n
1
1
n
n
bn =
and
are both increasing se
Homework 1 Math 118B, Winter 2010
Due on Thursday, January 14, 2010
1. Let f be dened for all real x, and suppose that M > 0 and > 0
such that
|f (x) f (y )| M |x y |1+ , x, y R.
Prove that f is constant.
2. Suppose that f (x) > 0 in (a, b). Prove that f
Homework 3 Math 118B, Winter 2010
Due on Thursday, February 4, 2010
1. Let be a xed increasing function on [a, b]. For u R(), dene
1/2
b
u
2
2
=
|u| d
.
a
Suppose f, g, h R(), and prove the triangle inequality
f h
2
f g
2
+ g h 2,
as a consequence of the
Self-Assessment Questions Math 118A, Fall
2009
1. For any sequence cfw_cn of positive numbers,
lim inf
n
cn+1
cn+1
lim inf n cn lim sup n cn lim sup
.
n
cn
cn
n
n
2. For any two real sequences cfw_an , cfw_bn , prove that
lim sup(an + bn ) lim sup an +
Homework 6 Math 118B, Winter 2010
Due on Thursday, March 11th, 2010
1. Let
0
sin2
fn (x) =
0
x
1
n+1
1
x < n+1 ,
1
x n,
1
x > n.
Show that cfw_fn converges to a continuous function, but not uniformly.
Use the series
fn to show that absolute convergence,
Homework 1 Math 118B, Winter 2010
Due on Thursday, January 14, 2010
1. Let f be dened for all real x, and suppose that M > 0 and > 0
such that
|f (x) f (y )| M |x y |1+ , x, y R.
Prove that f is constant.
Proof: If we divide by |x y |, we get
f (x) f (y )
HOMEWORK 10 FOR 18.100B AND 18.100C, FALL 2009
DUE FRIDAY, DECEMBER 4 AT NOON IN 2-108.
HW10.1 Rudin Chap 7, Prob 3: Construct sequences of functions cfw_fn , cfw_gn which
converge uniformly on some set E , but such that cfw_fn gn does not converge
unif
Excercise 1: Let A1 , . be subsets of a metric space. a) If Bn = A1 A2 . An , then prove Bn = n Ai i=1 b) If B = A1 ., then prove that Ai Bn i=1 c) Give an example where Ai = Bn i=1 Proof. a) Since each Ai is a closed set (see Theorem 2.27a page 35), and
Solutions to homework 7
By H akan Nordgren RUDIN PROBLEMS: Question 18 from page 168: Let (fn ) be a uniformly bounded sequence of Riemann integrable functions on [a, b]. Dene Fn (x) :=
x a
fn (t)dt,
for x [a, b]. Prove that (Fn ) has a uniformly converge
MATH 209: PROOF OF EXISTENCE / UNIQUENESS THEOREM FOR FIRST ORDER DIFFERENTIAL EQUATIONS
INSTRUCTOR: STEVEN MILLER Abstract. We highlight the proof of Theorem 2.8.1, the existence / uniqueness theorem for rst order dierential equations. In particular, we
Chapter 6 Riemann-Stieltjes Integral.
Subject: Real Analysis (Mathematics) Level: M.Sc.
Source: Syyed Gul Shah (Chairman, Department of Mathematics, US Sargodha)
Collected & Composed by: Atiq ur Rehman (atiq@mathcity.org), http:/www.mathcity.org
Introduc
Rudins Principles of Mathematical Analysis: Solutions to Selected Exercises
Sam Blinstein UCLA Department of Mathematics March 29, 2008
Contents
Chapter 1: The Real and Complex Number Systems Chapter 2: Basic Topology Chapter 3: Numerical Sequences and Se
Problem Set 5 Math 117, Winter 2010
Due on Tuesday, 02/16/2010, before class starts Section 11: 11.6 Section 12: 12.1, 12.3(e,h,l,n), 12.4(e,h,l,n), 12.11, and 12.15 Section 13: 13.2, 13.5, 13.6, 13.7
Basic Analysis
Introduction to Real Analysis
by Ji Lebl
r
December 23, 2009
2
A
Typeset in L TEX.
Copyright c 2009 Ji Lebl
r
This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0
United States License. To view a copy o
MINI SHOPAHOLIC
Cover
unamed
MINI SHOPAHOLIC
Sophie Kinsella
Contents
Cover
Title
Copyright
Dedication
Also by Sophie Kinsella
Tick Tock Playgroup
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chap
LECTURE NOTES ON RANDOM WALKS
IN RANDOM ENVIRONMENT
Ofer Zeitouni
Department of Electrical Engineering
Department of Mathematics
Technion Israel Institute of Technology
Haifa 32000, Israel
September 17, 2003. Version 3.1
Corrections inserted May 9, 2006.
Chapter I
Sums of Independent Random Variables
In one way or another, most probabilistic analysis entails the study of large
families of random variables. The key to such analysis is an understanding
of the relations among the family members; and of all t
Brownian Motion
Draft version of May 25, 2008
Peter Mrters and Yuval Peres
o
1
Contents
Foreword
7
List of frequently used notation
9
Chapter 0. Motivation
13
Chapter 1. Denition and rst properties of Brownian motion
1. Paul Lvys construction of Brownian
Steven E. Shreve
Stochastic Calculus for Finance I
Students Manual: Solutions to Selected
Exercises
December 14, 2004
Springer
Berlin Heidelberg NewYork
Hong Kong London
Milan Paris Tokyo
Contents
1
1
Probability Theory on Coin Toss Space . . . . . . . .