Need help with upper limit of sequence.

[c0dy]

Homework Statement

Prove that,
s^{*} = \lim_{n \rightarrow \infty} \sup_{k \geq n} s_k
Assume that s^{*} is finite.

Homework Equations

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

The Attempt at a Solution

I started out writing what I know.
By assuming s^{*} is finite, then \{s_k\} is bounded above so a supremum exists.
I'm unclear what exactly \sup_{k \geq n} s_k 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?

 

[c0dy]

Any hints?

 

[STEMucator]

\displaystyle\sup_{k ≥ n} s_k means that you want to take the supremum of the set s_k generated by all numbers which are greater than or equal to your n.

 

[c0dy]

\displaystyle\sup_{k ≥ n} s_k means that you want to take the supremum of the set s_k generated by all numbers which are greater than or equal to your n.

Ok that makes sense. Then from there I could say:

Let E = \{s_k\} ^{\infty}_{k=n}. Since s^* is finite, then E is bounded from above, E \subset \{s_k\} and E is not empty, then a supremum exists in E. And then taking limit as n \rightarrow \infty, E would consist of only supE which is exactly s^*