# Definitions, Proofs and Examples

This year (2011) I gave five optional examples classes to my second-year G12MAN Mathematical Analysis students on Definitions, Proofs and Examples.

I recorded videos of these classes, to go along with my previous videos on How and why we do mathematical proofs

These videos and associated materials are available below. They are also available for viewing or downloading from the University of Nottingham’s U-Now OER collection at

http://unow.nottingham.ac.uk/resources/resource.aspx?hid=966dbda8-3f2e-ae0b-f887-382f0ca6b716

These sessions are intended to reinforce material from lectures, while also providing more opportunities for students to hone their skills in a number of areas, including the following: working with formal definitions; making deductions from information given; writing relatively routine proofs; investigating the properties of examples; thinking up examples with specified combinations of properties.

The questions discussed were selected from the question sheet

More practice with de finitions, proofs and examples,

where you can also find details of some non-standard terminology used in this module.

Background required: a knowledge of the relevant definitions from Mathematical Analysis, especially in the setting of the real line, including the following: bounded sets; interior points and open sets; convergent sequences; continuous functions.

• Definitions, Proofs and Examples 1
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
Screencast
:
Streaming video (requires flash); mp4 file
• Definitions, Proofs and Examples 2
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
Screencast:
Streaming video (requires flash); mp4 file
• Definitions, Proofs and Examples 3
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.
Screencast: Streaming video (requires flash); mp4 file
• Definitions, Proofs and Examples 4
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.UnfortunatelyI failed to switch on the mains power, and so my tablet was running on battery power for this session, and (as usual) this has resulted in some synchronization problems between the andio and video. As a result, I am providing two versions of the video: in the second version the PiP footage of me stops after 2 minutes 47 seconds (when I start writing). Comments welcome!

Screencast (two versions)

• Definitions, Proofs and Examples 5
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.
Screencast: Streaming video (requires flash); mp4 file

These sessions now also have a YouTube PlayList at

This PlayList also includes the three earlier videos on How and why we do mathematical proofs

Similarly, if you have the free software iTunes, DPE has an iTunes album at
http://itunes.apple.com/gb/itunes-u/definitions-proofs-examples/id477470237

### 3 Responses to Definitions, Proofs and Examples

1. The audio on these “DPE” sessions was recorded using the webcam’s microphone, since this is very convenient to use. It’s not quite as good as the Sennheiser sound quality, of course, but it seems good enough to me given the additional fuss is saves.
I am quite pleased with the visual quality of the video, given that I have not been taking my desk lamp with me! It may be that I have just been lucky with the local lighting conditions, but so far I haven’t had anything close to the problems that I had when I was using older webcams.

2. There are some issues with the DPE4 recording. I suspect that I failed to turn the mains power switch on, and that I was operating on battery power. This lhas resulted in some additional distortion in the audio (but not too bad) and a slight loss of synchronization which becomes quite noticeable near the end.
I might recompile without the PiP of me in the corner, or with me there for just some of the time. I probably won’t have time to work through fixing the synchronization properly throughout.
I wonder, would there be interest in versions of my videos with no PiP?
There is also the option of compiling, flash-controlled, side-by-side videos where (I think) you have control over whether I am visble or not. I haven’t tried that one yet.
I may produce a few alternative versions and seek comments!