Show that the unit sphere with centre $0$ in $\mathbb{R}^d$ is compact.

Question

Namely, the sphere is $\{x\in\mathbb{R}^d: \| x\|_2=1\}$. I am going about this by proving that the sphere is bounded and closed. I have proved that it is bounded and I can see that it must be closed but I don't know how to write it out, can this be done without using the continuous map method?

Answer 1

$D=\mathbb{R}^d\setminus(B_1\cup B_2)$, where $B_1=\{x\in\mathbb{R}^d: \| x\|_2<1\}$ and $B_2=\{x\in\mathbb{R}^d: \| x\|_2>1\}$. Note $B_1,B_2$ are open. (You can find a proof about open ball is open here, and proof for $B_2$ is open is similar.) Hence $D$ is closed.