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 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.

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