Let

A be a bounded subset of R with infinitely many elements. If

s

= supA is NOT an element of A, prove that there exists a strictly

increasing sequence (xn), xn inA such that limit as n approaches infinity of (xn) goes to s.

I know how to prove that the limit of a strictly increasing sequence goes to the supremum, but I'm not sure how to show that there must exist a strictly increasing sequence if the supremum is not an element of the subset. Can someone show me how to do this? I would really really appreciate it.