## Proof involving least upper bound and unions

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

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