Tricki
a repository of mathematical know-how

When constructing sequences, keep track of the size of the terms (elementary real analysis)

When constructing sequences, keep track of the size of the terms (elementary real analysis)

The example that sticks in my mind is proving that k-cells [a,b]^k are compact. The proof is by contradiction: suppose that an open over has no finite subcover, and find a sequence of nested k-cells with no finite subcover. Eventually, one must be small enough to be covered with one neighborhood. When I first attempted the proof, I figured this much out for myself, but I didn't think to put a bound on the size of the terms and couldn't finish the proof. Now it seems like an obvious trick. (By bound, I mean set C=|b-a|, and make the n-th interval smaller than C/2^n.)

If this seems like a sufficiently general idea, I could write up this example. I think it needs a better title. I know I've used this trick for other problems, but I can't think of them off hand.

This proof could also be used as an example of using nested intervals (or nested compact sets), but I can think of examples where nested compact sets are used, but this bound trick isn't.