Every bounded sequence is Cauchy?

Question

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

Answer 1

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.