- #1

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## Homework Statement

Prove that,

[tex]s^{*} = \lim_{n \rightarrow \infty} \sup_{k \geq n} s_k[/tex]

Assume that [itex]s^{*}[/itex] is finite.

## Homework Equations

Definition of [itex]s^{*}[/itex] is here: http://i.imgur.com/AWfOW.png

## The Attempt at a Solution

I started out writing what I know.

By assuming [itex]s^{*}[/itex] is finite, then [itex]\{s_k\}[/itex] is bounded above so a supremum exists.

I'm unclear what exactly [tex]\sup_{k \geq n} s_k[/tex] means. Fixing n and finding supremum of {s_k} for k >= n and then letting n -> oo? I would think if there is an upper limit for {s_k} and for all n < k, as n ->oo then {s_n} will converge to that upper limit. And I have a feeling the Theorem 3.17 in the image might be applicable to this problem?