### Bolzano-Weierstrass Property

**Theorem 4.21 (Bolzano-Weierstrass Property)** A set of real numbers 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 .

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

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