Chapter 1
Axioms of the Real Number System
1.1
Introductory Remarks: What constitutes a proof ?
One of the hurdles for a student encountering a rigorous calculus course for the rst time,
is what level of detail is expected in a proof. If every statement i

Math 317 HW #5 Solutions
1. Exercise 2.4.2.
(a) Prove that the sequence defined by x1 = 3 and
xn+1 =
1
4 xn
converges.
Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show that
(xn ) is decreasing and bounded. To do so, I will pr

Analysis I
Math 413 Winter 2007
Professor Ben Richert
Exam 1
Solutions
Problem 1. (20pts) Definitions: state precisely each of the following. You should use the definition, and
not an equivalent statement.
(a10pts) If f : [a, b] R, and : [a, b] R is monot

UNIVERSITY EXAMINATIONS
UNISA
UNIVERSITEITSEKSAMENS
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m.“
MAT261 3 Maleune 2015
REAL ANALYSIS
Duaton 2 Hours 100 Marks
EXAMINERS :
FIRST : PROF A BATUBENGE—TSHIDIBI
SECOND ‘ DR L LINDEBOOM
Closed book examination.
Thls examination question paper remain

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
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57mm
3 MayIJune 2014
REAL ANALYSIS
Duration 2 Hours 100 Marks
EXAMINERS :
FIRST PROF A BATUBENGE-TSHIDIBI
SECOND DR L LINDEBOOM
Closed book examination.
This examination question paper remains the

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
UNISAIE
3 OctoberiNovember 2014
REAL ANALYSIS
Duration 2 Hours 100 Marks
EXAMINERS
FIRST PROF A BATUBENGE—TSHIDIBI
SECOND DR L LINDEBOOM
Closed book examination
TI‘IIS examination question paper remains the

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
UNISAE:
3 October/November 2013
REAL ANALYSlS
Durahon 2 Hours 80 Marks
EXAMINERS '
FIRST DR JRA GRAY
SECOND DR L LINDEBOOM
Closed book examination.
This examination question paper remains the property of th

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
UNISAE
MAT261 3 Maleune 2013
REAL ANALYSIS
Durahon 2 Hours 100 Marks
EXAMINERS :
FIRST DR JRA GRAY
SECOND DR L LINDEBOOM
Closed book examination.
This examination questlon paper remains the property of the

Introdution to Analysis
Irena Swanson
Reed College
Spring 2015
Table of contents
Preface
7
The briefest overview, motivation, notation
9
Chapter 1: How we will do mathematics
13
Section 1.1: Statements and proof methods
13
Section 1.2: Statements with qua

Introduction to Real Analysis
M361K
Preface
These notes are for the basic real analysis class. (The more advanced class is M365C.)
They were writtten, used, revised and revised again and again over the past ve years.
The course has been taught 12 times by

Induction Problems
Tom Davis
[email protected]
http:/www.geometer.org/mathcircles
November 7, 2005
All of the following problems should be proved by mathematical induction. The problems are not
necessarily arranged in order of increasing difficulty.

[Go to the Basic Analysis home page]
Basic Analysis: Introduction to Real Analysis:
Errata
This page lists errors in the various editions. Nonmathematical typos, misspellings, and grammar or style
problems are not listed here. Also not listed are things t

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
U N I SA
mm
3 OctoberfNovember 2015
REAL ANALYSIS
Durahon 2 Hours 100 Marks
EXAMINERS :
FIRST . PROF A BATUBENGE-TSHIDIBI
SECOND DR L LINDEBOOM
Closed book‘ examination
This examination question paper tem

REAL ANALYSIS NOTES
These notes are Copyright to Professor J K Langley and the University of Nottingham, but
are freely available for personal use.
Section 1. Introduction
This course is about functions of a real variable. Topics will include:
Review of S

4/5/05
Sample Paper Solutions
1. (a) First note that 0 x < y y x > 0. Also,
y n xn = (y x)(y n1 + y n2 x + y n3 x2 + + yxn2 + xn1 )
Now x 0 and y > 0 implies y nk1 xk 0 for all 1 k n 1, that y n1 > 0, hence
y n1 + y n2 x + y n3 x2 + + yxn2 + xn1 > 0 and y

SOLUTION OF HW1
MINGFENG ZHAO
September 10, 2012
1. [10 Points, Exercises 2, on Page 38] Suppose that xn is a sequence of real numbers that converges
to 1 as n . Using Definition 2.1, prove that each of the following limits exists.
a. 1 + 2xn 3 as n .
xn

COT3100 Discrete Structures and Applications- HomeWork 2 Solution 1
Problem 1 [Section 1.5 Exercise 14 (f)(g) (5 points)]
Use quantifiers and predicates with more than one variable to express these statements.
f) Some students in this class grew up in the

Carnegie Mellon University
Research Showcase @ CMU
Department of Philosophy
Dietrich College of Humanities and Social Sciences
1985
Logic and Argument Analysis: An Introduction to
Formal Logic and Philosophic Method
(REVISED)
Preston K. Covey
Carnegie Mel

Chapter 2
Sequences and Series
2.1 Sequences
A sequence is a function from the positive integers (possibly including 0) to the
reals. A typical example is an = 1/n defined for all integers n 1. The notation an
is different from the standard notation for f

An Interactive Introduction to
Mathematical Analysis
Jonathan Lewin
Junk Chapter
This is a junk chapter to force the table of contents to begin on page v.
Publisher, please throw out this page.
Publisher, please throw out this page.
Publisher, please thro

CHAPTER 3
Sequences and Series
3.1. Sequences and Their Limits
Definition (3.1.1). A sequence of real numbers (or a sequence in R) is a
function from N into R.
Notation.
(1) The values of X : N ! R are denoted as X(n) or xn, where X is the
sequence.
(2) (

Solutions to Practice Problems
Exercise 3.7
n
: n N.
Consider the set A = cfw_ (1)
n
(a) Show that A is bounded from above. Find the supremum. Is this supremum a maximum of A?
(b) Show that A is bounded from below. Find the infimum. Is this infimum
a mini

REAL ANALYSIS 2015/2016 (MATH20111)
MARCUS TRESSL
Lecture Notes
Homepage of the course:
http:/personalpages.manchester.ac.uk/staff/Marcus.Tressl/teaching/RealAnalysis/index.php
Contents
1.
Sequences
1.1. Definition and first examples of sequences
1.2. Con

3
The Limit of a Sequence
3.1
Definition of limit.
In Chapter 1 we discussed the limit of sequences that were monotone; this
restriction allowed some short-cuts and gave a quick introduction to the concept.
But many important sequences are not monotonenum

MAT2613/202/2/2016
Tutorial Letter 202/2/2016
REAL ANALYSIS
MAT2613
Semester 2
Department of Mathematical Sciences
This tutorial letter contains solutions
for assignment 02.
BAR CODE
Define tomorrow.
university
of south africa
Question 1
!
1
(1.1) (ar ) =

MAT 2613 SEMESTER 02
REAL ANALYSIS
GUIDE-LINES TO THE FINAL EXAMINATION
I n what follows we denote the text book Haggarty by H and the study guide by SG.
1. The assignment problems are fairly representative of the level and nature of the questions you
can