Bolzano-Weierstrass Theorem

Bolzano-Weierstrass Theorem: Every bounded, infinite set of real numbers has a limit point.

Lemma
Let S be a set with endowed with a linear order \leq such that for all nonempty X\subseteq S, both \inf(x) and \sup(x) exists and belong to S. Then S is finite.

Proof of the lemma: Suppose that S satisfies the linear order \leq and it is infinite. For n\geq0, we can recursively construct x_n:= \inf (S \backslash \{ x_i : i < n \} ) and X=\{ x_n, n\geq0 \}. Then (x_n) is an infinitely increasing sequence of member of S. Hence, \sup (X) \notin X which is a contradiction.

Proof of Theorem:
By contraposition. Let S be a bounded subset of \R and assume it has no limit point. Suppose X\subseteq S , as constructed above, is nonempty. Then \inf(X) \in X. Note that \inf(X) is a limit point of X, hence of S too. Analogously, \sup(X) \in X. Then by Lemma , S is finite. [1]