How to prove every closed interval in R is compact? Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familiar to us. Here I want to collect the ways to prove $[a,b]$ is compact.
Thanks for your help and any link.

Perhaps the shortest, slickest proof I know is by Real Induction.  See Theorem 17 in the aforelinked note, and observe that the proof is six lines.  More importantly, after you spend an hour or two familiarizing yourself with the formalism of Real Induction, the proof essentially writes itself.
