Real analysis: compact sets and intervals

Question

I am wondering whether the following statement is true or not.

Let $S\subseteq \mathbb{R}$ be a compact set such that $\{0,1\}\subseteq S$ and $S\subseteq [0,1]$. If $S\neq [0,1]$ then there exist two real numbers $0\le a<b\le 1$ such that $S\cap [a,b]=\{a,b\}$.

I feel that this should be true (I was not able to construct a counterexample), but I do not see how to prove it.

Answer 1

HINT: Suppose $x\in[0, 1]\setminus S$. Let

• $L=\{s\in S: s<x\}$, and

• $R=\{s\in S: s>x\}$.

Since $S$ is compact, $S$ is closed - so what do you know about $\sup L$ and $\inf R$?