*Part IB, Analysis II*

Quotations:

"The *f _{n}*'s are as nice as you can think of, but the limit is a terrible function.

"I won't let you go without letting you know what [uniform continuity] is. That would be cruel."

"Here is a definition of uniform convergence. Let me put you all to sleep."

"There are various *N*'s that will achieve very good things."

"There's a very cute way of doing this using the fundamental theorem of calculus..."

"At the very beginning, someone came and gave us an epsilon."

"Given *ε* > 0, there exists *N _{0}* such that all this happens."

"We're now going to leave the wonderful realm of uniform convergence..."

"Uniform continuity: It's going to look very much like continuity, but it's uniform."

"Our purpose from here until the end of the lecture will be to make this very small."

"This becomes a really fun game in analysis."

"And since you're in a generalising mood today..."

"There are two kinds of generalisations: ... "

"We'll see this in the next lecture... once I stop saying things I'm not supposed to." *after talking about topological spaces*

*Written on the blackboard:* "A convergent subsequence of a convergent sequence is convergent."

"Now it's all been served in a tray, what can I do to finish the proof?"

"And now a theorem that really doesn't deserve to be called a theorem."

"For proximity, any balls will do."

*Also written on the board:* "Lemma 2: An open ball is open. A closed ball is closed."