Let S be a non-empty set of real numbers and suppose that L is the least upper bound for S. Prove that there is a sequence of points x_n in S such that x_n\rightarrow L as n\rightarrow\infty.

Prove or disprove the following:
For non-empty bounded sets S and T: \sup(S\cup T)=\max\{ \sup(S), \sup(T)\}
(I had to use sup instead of lub, which was what was originally written)