Question

Let (xn) be a bounded real sequence. Let
yn = sup{xm: m>=n} = sup{xn, xn+1, xn+2, ...}.Prove that the sequence (yn) converges.

Answer 1

I'd prove this by showing that the sequence (yn) is Cauchy. (Because then we can rely on the fact that every Cauchy sequence in the real numbers converges in the reals.

To show (yn) is Cauchy I would formalize with epsilon-N the following idea: Suppose (yn) weren't Cauchy. Then it would mean that for every k > M some M, there is a j(k) > k such that |y_k - y_j(k) | is larger than any given epsilon ... which means that we are removing some subsequence of (xk) from the sequence (xn) and the subsequence has the property that it is strictly decreasing and the gap between successive elements of the sub-sequence is bounded below. And hence the sequence itself must trend to -infinity. But that's a violation of the condition that (xn) is bounded.