### Bolzano-Weierstrass Property

Theorem 4.21 (Bolzano-Weierstrass Property) A set of real numbers $\mathrm{E}$ is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to $\mathrm{E}$.

Proof Suppose that $\mathrm{E}$ is both closed and bounded and let $\mathrm{\{x_n\}}$ be a sequence of points chosen from $\mathrm{E}$. Since $\mathrm{E}$ is bounded this sequence $\mathrm{\{x_n\}}$ must be bounded too. We apply the Bolzano-Weierstrass theorem for sequences (Theorem 2.40) to obtain a subsequence $\mathrm{\{x_{n_k}\}}$ that converges. If $\mathrm{x_{n_k}\rightarrow z}$ then since all the points of the subsequence $\mathrm{\{x_{n_k}\}}$ belong to $\mathrm{E}$ either the sequence is constant after some term or else $\mathrm{z}$ is a point of accumulation of $\mathrm{E}$. In either case we see that $\mathrm{z \in E}$. This proves the theorem in one direction.