## Mathematics: Definitions, Proofs and Examples

**Mathematics: Definitions, Proofs and Examples (University of Nottingham)**. This is a series of video lectures given by Dr. Joel Feinstein.
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.

**Lecture 1 - Why do we do proofs?**

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.

**Lecture 2 - How do we do proofs? Part I**

The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. Part I is suitable for anyone with a knowledge of elementary algebra and functions.

**Lecture 3 - How do we do proofs? Part II**

The aim of these sessions on how we do proofs is to help students with some of the relatively routine aspects of doing proofs. Part II requires some background knowledge of convergence and divergence of series of real numbers.

**Lecture 4 - 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.

**Lecture 5 - 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.

**Lecture 6 - 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.

**Lecture 7 - 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.

**Lecture 8 - 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.