Proofs: You will be asked to do some proofs on the midterm. The proofs will be chosen from the following:
- Show that an arbitrary union and a finite intersection of open sets are open
- Show that an arbitrary intersection and a finite union of closed sets are closed.
- Prove (using the sequence definition) that if f and g are continuous, then fg, f+g, f(g) (ie composition) are all continuous.
- Fixed Point Theorem: Let f(x) be continuous and x_{n+1} = f(x_{n}) is a sequence produced by iteration. If x_{n} converges to a limit l, then l is a fixed point of f.