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.

If you're wanting a proof, take the following.
Let $\mathscr{U}$ be an open cover of $[a,b]$ and let $$b'=\sup\{x\in[a,b]:[a,x]\text{ is covered by finitely many elements of the open cover}\}.$$ Clearly $b'\leq b$. Now, if $b'<b$, then it's contained in some open set from the cover implying $b'$ is contained in some open interval. This shows that there is an element greater than $b'$, say $b''\in[a,b]$ that is covered by finitely many open sets from the cover, a contradiction to $b'$ being the supremum. Hence $b'=b$ and $[a,b]$ is compact.
