Is a Monotone Sequence with a Bounded Subsequence Always Bounded?

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Homework Help Overview

The discussion revolves around the properties of monotone sequences in relation to bounded subsequences, specifically addressing whether a monotone sequence that contains a bounded subsequence is itself bounded.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the definition of monotone sequences and the implications of having a bounded subsequence. There are attempts to connect the convergence of subsequences to the boundedness of the entire sequence.

Discussion Status

Some participants are exploring the relationship between convergent subsequences and the overall sequence, with one suggesting the use of the supremum of the limits of convergent subsequences as a potential approach. There is an ongoing examination of the implications of these ideas without a clear consensus yet.

Contextual Notes

Participants are considering the definitions and properties of monotone sequences and boundedness, as well as the potential complexity introduced by multiple convergent subsequences.

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Homework Statement



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

Homework Equations





The Attempt at a Solution

 
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did you even try to solve this?
 
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
 
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.
 

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