Prove or give a counter example:
A) Every bounded sequence has a Cauchy subsequence
B) Every monotone sequence has a bounded subsequence
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.