These web pages present a web-based first course in real analysis
(part of what is covered at university level in the USA as
calculus

though this terminology does not apply in the UK and the
notes here follow the UK style). In particular these pages cover the
notions of convergence of sequences and series and the nature of the
real numbers. Content is provided by a mixture of HTML and
MathML. All content here may be copied as often as you like under the
terms of the Gnu Free
Documentation Licence.

These web pages were originally written to support the modules MSM1Bb and MSM2P01b given in the School of Mathematics, University of Birmingham, from 2005-6 onwards, for students taking single honours or combined/joint honours programmes in Mathematics. They have since been updated for subsequent years, and now contain rather more material than is required for this module, or other similar modules.

The pages will form a useful or main source of information for students taking this module or a similar module and may be of interest to many other people too for mathematical reasons. They also provide a working example of the Gloss system for authoring HTML and MathML. The pages were updated recently (2015) to take advantage of MathJax and a new version of Gloss.

The author is Dr Richard Kaye, http://web.mat.bham.ac.uk/R.W.Kaye/ Email: R.W.Kaye-at-bham.ac.uk.

All of the web pages with URI starting

They may be copied under the terms of the
Gnu Free Documentation Licence.
This means that you may download, copy and modify them at
will, and make your own private version. If you distribute this
private version or indeed any work at all derived from these pages
you *must* distribute it under an identical or compatible licence.
In particular you may not charge for your work save a reasonable charge
for transmission costs or electronic media.

Almost all these pages contain mathematics in MathML format, and are rendered in your browser using javascript. More details are here.

Please email me at `R.W.Kaye[at]bham.ac.uk` if you find these
notes useful, or have any other comment on them.

The mathematical content in these pages is a typical
first course in real analysis of the type usually given in
first or second year at a British university. The main
idea being developed here is the idea of a *limit*
of a sequence of numbers or of an infinite series of numbers.
We can think of a sequence or series as an infinite process
that may (or may not) converge to a value, and the first
step is to define the correct notion of convergence.
The ideas are particularly rich and there are many things
one can say about convergence in general, and some striking
examples. A related subsidiary idea in these notes is
the nature of the real numbers themselves, because
these numbers can be characterised exactly by the way
they behave under the concept of limit. It is often a
surprise to beginners to the subject that the main concept in
mathematics - number - is secondary to a somewhat trick topic in
first year analysis.

As a result, the ideas of convergence studied here will be enough to make precise the nature of the real numbers, integration, differentiation, and many functions such as exponentiation and trigonometric functions, all of which need the ideas of convergence and limits to be defined precisely. (It is often a surprise to students that the definitions of these apparently familiar ideas require such sophisticated work.)

Because it is easy to make logical errors and fall into
paradoxes

it is important to be very precise and rigorous
in doing this work. In fact, a major part of these notes is
an introduction to mathematical rigour and abstract
mathematics, and this in any case forms an important part of any
first-year pure mathematics course. So these notes also
include a certain amount of material about logic, proofs and quantifiers,
much of which is unfamiliar to students beginning this topic.

Preamble: In the following Licence Notice, this document

refers to all web pages and other electronic documents available via
http at any of the addresses
**There is no warranty of any kind on these web pages.**

*Copyright (c), 2006,2015, Richard Kaye,
http://web.mat.bham.ac.uk/R.W.Kaye/.
Permission is granted to copy, distribute and/or modify this document
under the terms of the
GNU Free Documentation License, Version 1.2
at http://www.gnu.org/copyleft/fdl.html
or any later version published by the Free Software Foundation; with
no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.*

When you have read the information above, including the licence, and are ready to do so, please click the following link to take you to the Table of Contents.