Every sequence in a closed and bounded set S in sequence Rn has a convergent subsequence (which converges to a point in S).

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Proof: Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set because the set is closed.

Conversely, every bounded sequence is in a closed and bounded set, so it has a convergent subsequence.

AnotherBolzano-Weierstrasstheorem is:

Every bounded infinite set of real numbers has at least one limit point or cluster point.

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