# Basic Analysis - Proof Bolzano Wierestrass by Least Upper Bound

## Homework Statement

Let (an) be a boundedd sequence, and define the set
$S= {x\in R : x < a_n$ for infinitely many terms $a_n\}$
Show that there exists a subsequence $(a_n_k)$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 $\epsilon > 0$ there are finitely many terms $a_n$ s.t. $a_n> s+\epsilon$
I thought of defining the subsequence to be $a_n_k = min(a_n | a_n > s+\frac{1}{k})$ 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!