Let $C\subset \mathbb{R}$ be compact. I am wondering if
$C=\bigcup_{i=1}^n[a_i,b_i]$
then for some $a_i,b_i\in\mathbb{R}$, $a_1\le b_1 < a_2 \le b_2 \dots < a_n \le b_n$. By Heine-Borel, $C$ does indeed lie in some interval $[a,b]$, but is it the finite disjoint union of such intervals?
Could not find a proof for myself yet. Perhaps this is even wrong?
Thank you in advance :)