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.

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, but the notion of uniform convergence is more subtle. Uniform convergence is explained in terms of closed function balls and Dr Feinstein's notion of sets absorbing sequences.

The differences between the two types of convergence are illustrated with several examples. Some standard facts are also discussed: a uniform limit of continuous functions must be continuous; a uniform limit of bounded functions must be bounded; a uniform limit of unbounded functions must be unbounded.

Target audience: Most of this material should be accessible to anyone who understands what a real-valued function is, and understands the notion of convergence of a sequence of real numbers. This should include most mathematics undergraduates by the end of their first year. An understanding of continuity and of boundedness for real-valued functions defined on various types of domain would help the student to understand the latter part of the material.

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 open sets are still open, but that infinite intersections of open sets need not be open any more.

See also Dr Feinstein's blog, Explaining Mathematics, http//explainingmaths.wordpress.com/

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 includes partitions of intervals; Riemann lower and upper sums (approximation sing rectangles); the Riemann lower and upper integrals; Riemann integrability of functions, and the Riemann integral. Examples are given of functions which are/are not Riemann integrable in particular, continuous real-valued functions on closed intervals are Riemann integrable. The lecture concludes with the statements of the (first) Fundamental Theorem of Calculus and the Mean Value Theorem of Integral Calculus.

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).

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 divergence of series of real numbers. A revision sheet covering relevant material is available at http://www.maths.nott.ac.uk/personal/jff/Teaching/Standard-Series.pdf

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 there are positive integers, as is shown in the more challenging final section, using Cantor's diagonalization argument. This last part of the talk is relatively technical, and is probably best suited to first-year mathematics undergraduates, or advanced maths A level students. Others may find the technical details hard to follow, and should focus on the overview. Dr Joel Feinstein's blog is available at http://explainingmaths.wordpress.com/

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/

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 divergence of series of real numbers. A revision sheet covering relevant material is available at http://www.maths.nott.ac.uk/personal/jff/Teaching/Standard-Series.pdf

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/