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 familar to us. Here I want to collect the ways to prove $[a,b]$ is compact.

But now, being a mature mathematician, when I am asked about the proof, I answer the following. Clearly, a two-point set $\{0,1\}$ is compact. Tychonov theorem implies that
Cantor set $\{0,1\}^\omega$ is compact too. At last, a segment is compact as a continuous image of Cantor set. 🙂