Prove or give a counter example:

A) Every bounded sequence has a Cauchy subsequence

B) Every monotone sequence has a bounded subsequence

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- Apr 1st 2007, 10:34 AMlearn18prove/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 - Apr 1st 2007, 10:39 AMThePerfectHacker
Every Bounded Sequence has a convergent subsequence.

(See Bolzano-Weierstrass Theorem).

But a convergent sequence is a Cauchy sequence.

Quote:

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.