# Math Help - Prove there is a subsequence of S, where L = supS, S nonempty + bdd abv.

1. ## Prove there is a subsequence of S, where L = supS, S nonempty + bdd abv.

Let S be a nonempty set that is bbd abv. Let u be the supremum of S. Show that there is a sequence ${x_n}$ in S with ${x_n}$ approaching u.
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I have two general ideas on how to solve this problem, but I don't know how to start.

1) I use the epsilon definition of convergence, defining epsilon with relation to u.

or

2) Show that the limit cannot be less than supS (easy to prove, or not? if lim < sup S, lim is in S, choose an increasing sequence and ${x_k}$ > lim), or greater than supS.

If not(greater or less than), must be equal to.
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Let S be a nonempty set that is bbd abv. Let u be the supremum of S. Show that there is a sequence ${x_n}$ in S with ${x_n}$ approaching u.
There are two cases to consider:
• 1. $u\in S$
• 2. $u\notin S$