Bolzano-Weierstrass_Theorem

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Definition

Every bounded sequence contains a convergent subsequence.

Proof

Let $(a_n)$ be a bounded sequence. Then there exists $M > 0$ such that $|a_n| \leq M$ for all $n$.

By the Nested Interval Property, we can construct a nested sequence of closed intervals $I_1 \supseteq I_2 \supseteq \cdots$ where each $I_k$ has length $M/2^{k-1}$ and contains infinitely many terms of $(a_n)$.

By repeatedly selecting terms from these intervals, we obtain a subsequence $(a_{n_k})$ that is Cauchy, and hence convergent by the Cauchy_Criterion.