Analysis 3 Proofs and Definitions Weeks 5 and 6

What is Bolzanno Weierstrass in n dimensions? Prove it

How is continuity defined in n dimensions?

What is the sequential definition of continuity in n dimensions?

Prove that the sequential and epsilon definition of continuity are equivalent

Definition of continuous limit

What is separate continuity?

Properties of continuous functions

Prove that the composition of continuous functions is continuous

Prove that vector projection is continuous

Complicated proposition about composition of continuous functions being continuous

What is continuity along lines or linear continuity?

Prove that continuity implies continuity along lines but not converse

What is a closed and open set in n dimensions?

Prove that the empty set is both closed and empty and so is R^n

An open set is open if and only if its complement is closed

What is the open and closed euclidean ball?

Prove that the open euclidean ball is an open set

Prove that the union of open sets is open

The epsilon neighbourhood around an open set is open

Finite intersection of open sets is also open

An arbitrary intersection of closed sets is closed and a finite union of closed sets is closed

What is topology then?

What does it mean to be continuous in terms of open sets

Prove and state the theorem of continuity via open and closed sets