Part IB, Analysis II
"The fn'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 N0 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."