Is a Monotone Sequence with a Bounded Subsequence Always Bounded?

[zachsdado]

Homework Statement

prove that a monotone sequence which has a bounded subsequence is bounded

Homework Equations

The Attempt at a Solution

 

[sprint]

did you even try to solve this?

 

[zachsdado]

Of course

I tried using the def. of a monotone sequence to show that the subsequence was monotone and bounded hence it converged to some number and then tried to prove that the sequence was convergent thus it was bounded

 

[ZioX]

The problem is of course that there could be a lot of convergent subsequences. Try taking the sup of the limits over all of the convergent subsequences. Claim: the sequence converges to this sup.

I think this should do it.