INTRODUCTION
TO REAL ANALYSIS
William F. Trench
Professor Emeritus
Trinity University
San Antonio, TX, USA
2003 William F. Trench, all rights reserved
Library of Congress Cataloging-in-Publication Data
Trench, William F.
Introduction to real analysis / Wi

MA 242
September 14, 2005
The vector space Rn
Denition. The vector space Rn is the set of all n-tuples of real numbers. That is, Rn is
the set of all possible n 1 column vectors of the form
x1
x2
.
.
.
xn
,
where xk is a real number for k = 1, 2, . . . ,

Foundations of Mathematics I
Set Theory
(only a draft)
Ali Nesin
Mathematics Department
Istanbul Bilgi University
Kutepe Sili
s
s
Istanbul Turkey
anesin@bilgi.edu.tr
February 12, 2004
2
Contents
I
Naive Set Theory
7
1 Basic Concepts and Examples
1.1 Sets,

2
Polynomials over a eld
A polynomial over a eld F is a sequence
(a0 , a1 , a2 , . . . , an , . . .)
where
ai F i
with ai = 0 from some point on. ai is called the ith coecient of f .
We dene three special polynomials. . .
0 = (0, 0, 0, . . .)
1 = (1, 0, 0

A First Course
in Topology
Common}
and Dimension
Iohn McClL-ar',’ Chapter 2
Metric and Tupnlagical
Spaces
Topology beg-431.5 when am am t'mpipmnini with
mm mhruiw pram-Hie: mam-rig mt: Ea dyﬁne
mnﬁntritnl.
Summon LEFSE'HETZ
In order tn fame a. fang-nag;-

Analysis Notes
(only a draft, and the rst one!)
Ali Nesin
Mathematics Department
Istanbul Bilgi University
Kutepe Sili
s
s
Istanbul Turkey
anesin@bilgi.edu.tr
June 22, 2004
2
Contents
1 Preliminaries
1.1 Binary Operation . . . . . . . . . . . . . . . . .

Basic Algebra
(only a draft)
Ali Nesin
Mathematics Department
Istanbul Bilgi University
Kutepe Sili
s
s
Istanbul Turkey
anesin@bilgi.edu.tr
February 12, 2004
2
Contents
I
Basic Group Theory
7
1 Denition and Examples of Groups
1.1 Denition and Basics . . .