- #1

mynameisfunk

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## Homework Statement

Suppose that {[itex]a_n[/itex]} is a bounded sequence of real numbers such that, for all [itex]n[/itex], [itex]a_n \leq \frac{a_{n-1}+a_{n+1}}{2}[/itex]. Show that [itex]b_n=a_{n+1}-a_n[/itex] is an increasing sequence. Otherwise show that {[itex]a_n[/itex]} converges.

## Homework Equations

## The Attempt at a Solution

I do not see it at all..

It seems that since [itex]2a_n \leq a_{n-1}+a_{n+1}[/itex] that the sequence could be either increasing OR decreasing. i could fit the set of integers, either increasing or decreasing and the inequality holds and [itex]b_n[/itex] is stagnate. BUT, since {[itex]a_n[/itex]} is bounded, it must be convergent?? i couldn't find any theorems or anything to support this claim though. Possibly "For a real valued sequence {[itex]s_n[/itex]}, [itex]\lim_{n \rightarrow \infty}=s[/itex] iff it's lim sup=lim inf =s (as n approaches infinity)" ??

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