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## Homework Statement

Let (a

_{n}) be a boundedd sequence, and define the set

[itex]S= {x\in R : x < a_n [/itex] for infinitely many terms [itex]a_n\}[/itex]

Show that there exists a subsequence [itex](a_n_k)[/itex]converging to s = sup S

## Homework Equations

This is supposed to be a direct proof of BW using the LUB property, so no monotonic convergence, Cauchy criterion, nested interval property etc...

## The Attempt at a Solution

I am having trouble thinking of a way to define the subsequence. What I can show is that, if [itex]\epsilon > 0[/itex] there are finitely many terms [itex] a_n [/itex] s.t. [itex]a_n> s+\epsilon[/itex]

I thought of defining the subsequence to be [itex]a_n_k = min(a_n | a_n > s+\frac{1}{k})[/itex] But I was having trouble proving that this subsequence converges to s. I would greatly appreciate a tip or prod in the right direction of defining a subsequence that will work. Thank you!