Let C be a compact set and assume $C \subset O$, where $O$ is an open set. Then, for any $x \in C$, there is an $r_x >0$ such that $B(x, r_x) = (x-r_x, x+r_x) \subset O$. Show that there is an $R >0$ such that $d(x , C) < R$ implies $x \in O$. Hint: use the Heine-Borel theorem.
The theorem says that for $[a,b] \in \mathbb{R}$, suppose that there exists a collection of open intervals that covers $[a,b]$. Then, there exists a union of finite number of collection of open intervals which contains $[a,b]$.
I guess $C$ is covered by a finite collection of open intervals. Then, since intervals have finite distance(?), distance is bounded by $R$.
I am new to this topic, so I am not sure how to do this. Could you give some hint?
Edit: $C \subset \mathbb{R}$ and I define a compact set to be a closed and bounded set.
Hint: First, let $s_x:=\frac 1 2 r_x$ and choose finitely many of the intervals $(x−s_x,x+s_x)$ that cover $C$. Let $R$ be the minimum of these values of $s_x$.