This is the first of three sessions by Dr Joel Feinstein on how and why we do proofs. Dr Feinstein's blog is available at http://explainingmaths.wordpress.com/ The aim of this session is to motivate students to understand why we might want to do proofs, why proofs are important, and how they can help us. In particular, the student will learn the following: proofs can help you to really see WHY a result is true; problems that are easy to state can be hard to solve (Fermat's Last Theorem); so

Level 2 Mathematics

Why do we do proofs? - (iPod) Dr Joel Feinstein

This video is a combination of the three screencasts from Chapter 9 of the second year module G12MAN Mathematical Analysis, lectured by Dr J. Feinstein (Nottingham). See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ and, in particular, the associated blog post at http://wp.me/posHB-7j The aim of this material is to introduce the student to two notions of convergence for sequences of real-valued functions. The notion of pointwise convergence is relatively straightforward, bu

Uniform convergence and Pointwise convergence (iPod) - Dr Joel Feinstein

This is a lecture on the properties of open sets in finite-dimensional Euclidean space by Dr Joel Feinstein from his second-year module on Mathematical Analysis. This material is suitable for those who already know the definitions of open set and of the interior of a set in finite-dimensional Euclidean space. In this session Dr Feinstein shows that finite unions and intersections of open sets are open, and then discusses infinite unions and intersections. It turns out that infinite unions of o

Properties of open sets - (iPod) Dr Joel Feinstein

This is the final lecture of the second-year module G12MAN Mathematical Analysis, as taught by Dr Joel Feinstein. This lecture gives a brief introduction to Riemann integration. This material is motivated in terms of questions of antidifferentiation and area. The proofs of the lemmas and theorems are not included here (see books for details), but the main definitions are given in full, along with illustrative examples and diagrams, and the statements of the main theorems. Material discussed inclu

An introduction to Riemann integration - (iPod) Dr Joel Feinstein

This is the first of two sessions on how to do proofs. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more cre

How do we do proofs? Part I (iPod) - Dr Joel Feinstein

The aim of the two sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs. Part II requires some background knowledge of convergence and dive

How do we do proofs? Part II (iPod) - Dr Joel Feinstein

This popular maths talk by Dr Joel Feinstein gives an introduction to various different kinds of infinity, both countable and uncountable. These concepts are illustrated in a somewhat informal way using the notion of Hilbert's infinite hotel. In this talk, the hotel manager tries to fit various infinite collections of guests into the hotel. The students should learn that many apparently different types of infinity are really the same size. However, there are genuinely "more" real numbers than the

Beyond Infinity? (iPod) Dr Joel Feinstein

In this workshop, Dr Feinstein helps students to understand convergence of sequences in finite-dimensional Euclidean space using his terminology of sets absorbing sequences (absorption). For more on the advantages of this terminology, see Dr Feinstein's blog post at http//wp.me/posHB-c See also Dr Feinstein's blog, Explaining Mathematics, http//explainingmaths.wordpress.com/

Sequence convergence and absorption - (iPod) Dr Joel Feinstein

The aim of the two sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs. Part II requires some background knowledge of convergence and diverge

How do we do proofs? Part II - Dr Joel Feinstein

Sequence convergence and absorption - Dr Joel Feinstein

Why do we do proofs? - Dr Joel Feinstein

Beyond Infinity? Dr Joel Feinstein

An introduction to Riemann integration - Dr Joel Feinstein

This is the first of two sessions on how to do proofs. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. In particular, we focus on how to start proofs, and how and when to use definitions and known results. With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more cr

How do we do proofs? Part I - Dr Joel Feinstein

Uniform convergence and Pointwise convergence

A selection of recordings suitable for second-year mathematics undergraduate students at university.