The "home" for these pages on Sequences and Series
is at http://web.mat.bham.ac.uk/R.W.Kaye/seqser/. You will find there
a preface

, as well as legal details (such as the rules you must follow
if you want to copy any part of this text).

- Home page and preface.
- Contents, this page.
- The MathML used in these pages and notes for setting up your web browser if required.

- Sequences and series: an introduction.
- The triangle inequality.
- Why do we need to do analysis rigorously?
- The definition of convergence.
- Logic for analysis: statements.
- Logic for analysis: proofs.
- Examples of convergent sequences.
- Sequences that are eventually constant.
- Theorem on uniqueness of limits.
- Theorems on bounds.
- Subsequences.
- Using subsequences to prove convergence.
- Sums of sequences.
- Products of sequences.
- Quotients of sequences.
- Continuity.
- Archimedean Ordered Fields.
- Density results for the rationals.
- Properties of monotone sequences.
- The Monotone Convergence Theorem and Completeness of the Reals.
- Applications of monotone convergence to roots.
- The arithmetic-mean/geometric-mean inequality.
- The Euler number e.
- The natural logarithm.
- The Bolzano-Weierstrass Theorem.
- Cauchy Sequences.
- Supremum and infimum.
- Fekete's lemma.

- Introduction to constructions of number systems.
- Construction of the natural numbers.
- Construction of the integers.
- Construction of the rationals.
- Construction of the reals via monotonic sequences.
- Construction of the reals via Cauchy sequences.
- Construction of the reals via Dedekind cuts.
- Construction of the reals via almost linear maps.
- The uniqueness of the reals.