I’m fortunate this week to have another topic suggested again by Mr Wu, blogger and Singaporean mathematics tutor. It’s a big field, so forgive me not explaining the entire subject.
Analysis is about proving why the rest of mathematics works. It’s a hard field. My experience, a typical one, included crashing against real analysis as an undergraduate and again as a graduate student. It turns out mathematics works by throwing a lot of $latex epsilon $ symbols around.
Let me give an example. If you read pop mathematics blogs you know about the number represented by $latex 0.999999cdots $. You’ve seen proofs, some of them even convincing, that this number equals 1. Not a tiny bit less than 1, but exactly 1. Here’s a real-analysis treatment. And — I may regret this — I recommend you don’t read it. Not closely, at least. Instead, look at its shape. Look…
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