Other proof for existence of monotone subsequences

Question

Is there any other proof of Bolzano-Weierstrass theorem (i.e.: Let ${\{x_n}\}$ be an arbitrary sequence of real numbers. Then ${\{x_n}\}$ has a monotone subsequence.), WITHOUT using concept of "peaks".

Thank you.

Answer 1

I would usually state the Bolzano-Weierstrass theorem as: any bounded sequence in $\mathbb{R}^N$ has a convergent subsequence. The proof for general $N$ follows by induction once you have the base case, which aside from the usual proof can be shown by showing that the limit superior of a sequence exists and that, for every $\epsilon>0$, the sequence contains a point at most $\epsilon$ distance from the limit superior. You could also use the construction of the limit superior to show the existence of a monotone subsequence, but this is pretty much the concept of peaks anyway.