Every sequence in a closed and bounded collection S in succession Rn has actually a convergent subsequence (which converges come a suggest in S).

You are watching: Every bounded sequence has a convergent subsequence

Proof: Every succession in a closed and also bounded subset is bounded, so it has a convergent subsequence, which converges come a allude 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.

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Additional Information

AnotherBolzano-Weierstrasstheorem is:

Every bounded infinite set of genuine numbers contends least one limit point or cluster point.


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