Gabriel Paternain

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." 1