The set $$\{(a,b,c,d)\in \mathbb R^4:a^2+b^2+c^2+d^2=1\}$$ is compact.
I want to know why this is true (or maybe, why this isn't, if that's the case).
When doing olympiad inequality problems, we can sometimes use the fact that continuous functions over compact sets attain a global minima/maxima. Since $\mathbb R^n$ isn't compact, we can often impose some conditions (e.g. $a^2+b^2+c^2+d^2=1$) to make the domain of the function compact. In olympiads we can make this claim without any proof. But due to my interest, I was interested in proving this rigorously. However, I couldn't do it despite having some background in analysis. Hence, I am asking this question here.
(By the way, this is my first post on this site. I couldn't find this by searching so I decided to ask it myself. If there's anything else I should be adding to my question, please tell me so. I hope I can become a useful contributor to this site.)