I know that every Cauchy sequence is bounded, but is the reverse true?
2026-03-29 20:47:03.1774817223
Every bounded sequence is Cauchy?
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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.