Every Bounded Sequence has a convergent subsequence.

(See Bolzano-Weierstrass Theorem).

But a convergent sequence is a Cauchy sequence.

No consider,B) Every monotone sequence has a bounded subsequence

s_n=n

We see that every subsequence must diverge to +oo.

Hence all sequences diverge to +oo.

(Excercise).

But then no subsequence is bounded because it diverges to +oo.