Thread: prove/counterexample for sequences

Prove or give a counter example:

A) Every bounded sequence has a Cauchy subsequence
B) Every monotone sequence has a bounded subsequence

Originally Posted by learn18

Prove or give a counter example:

A) Every bounded sequence has a Cauchy subsequence

Every Bounded Sequence has a convergent subsequence.
(See Bolzano-Weierstrass Theorem).
But a convergent sequence is a Cauchy sequence.

B) Every monotone sequence has a bounded subsequence

No consider,
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.