Definitions, proofs and examples

An easy proof by contradiction concerning sets absorbing sequences; a proof that various statements about convergence of sequences in a non-empty set are equivalent to the set having exactly one point; various examples relating to (non) sequential compactness and divergence of subsequences.

Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.

A close look at sequences of real numbers which tend to plus or minus infinity, and connections with the (non)-existence of bounded subsequences and/or convergent subsequences.

Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.

How do we do proofs? (Part II)

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 creative and interesting aspects of constructing proofs.

Part I is suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from

the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions).

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); sometimes statements which appear to be intuitively obvious may turn out to be false (Simpson's paradox); the answer to a question will often depend crucially on the definitions you are working with. Target audience: suitable for anyone with a knowledge of elementary algebra and prime numbers, as may be obtained by studying A level mathematics.

Discussion of questions relating to: unions of finite sets, bounded sets and closed sets; convergence of sequences, and the related (non-standard) concept of absorption of sequences by sets.

Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.

Discussion of questions relating to: Cartesian products, set differences and set inclusions; bounded sets and unbounded sets; open sets and sets which are not open; continuous functions, divergent sequences and convergent sequences.

Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.

Discussion of questions relating to: set inclusions and set equalities; sums of subsets of the real line; examples showing the difference between sum and union.

Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.