Let S be a nonempty bounded above subset of R . prove that there is a sequence $\displaystyle x_n \subseteq S$ of elements of S that converges to sup S
Let $\displaystyle s=\sup S.$ Then $\displaystyle \forall\,n\in\mathbb N,$ $\displaystyle \exists\,x_n\in S$ such that $\displaystyle s\geqslant x_n>s-\frac1n$ since $\displaystyle s-\frac1n$ is not an upper bound for $\displaystyle S.$ The sequence $\displaystyle \left(x_n\right)_{n\,=\,1}^\infty$ converges to $\displaystyle s.$