How to prove every closed interval in R is compact?

Let [a,b]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.

Thanks for your help and any link.

Answer

As I remember, a long time ago, when I was a student, at an exam my teacher asked me to prove this. I answered to him, that this result is so simple, that I forgot how to prove it. 🙂

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}ω is compact too. At last, a segment is compact as a continuous image of Cantor set. 🙂

Attribution
Source : Link , Question Author : Paul , Answer Author : Alex Ravsky

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