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 $\displaystyle x_n$ in S such that $\displaystyle x_n$ --> L as n --> $\displaystyle \infty$.

Prove or disprove:

For non-empty bounded sets S and T: lub(SUT) = max{lub(S), lub(T)}.