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I know that every Cauchy sequence is bounded, but is the reverse true?

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  • $\begingroup$ Whether every bounded sequence has Cauchy subsequence is a bit more interesting question. $\endgroup$ Nov 25, 2016 at 18:04

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No. Consider the sequence $$1,-1,1,-1,1,-1,\dots$$ Clearly this seqeunce is bounded but it is not Cauchy. You can show this directly from the definition of Cauchy.

Alternatively, every Cauchy sequence (in $\mathbb{R}$) is convergent. Clearly the above sequence is not, thus it is not Cauchy.

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