Smallest closed interval containing $E$

Question

Let $E$ be a closed set. Suppose $[c, d]$ is the smallest closed interval containing $E$. Prove that $c\in E$ and $d\in E$.

This is actually part of a bigger proof - that a continuous function achieves an absolute maximum and absolute minimum on a compact interval. I'm sure this is an easy one but a hint would be much appreciated.

Answer 1

Hint: closed sets contain their limit points. Can you construct sequences converging to $c$ and $d$ respectively.

Answer 2

Hint: Begin your proof with: “If $c\notin E$, then since $E^\complement$ is open,  …”