Proving a set is a Borel Set

Question

I understand every open set (and closed for that matter) is a Borel set. The thing is, I've been given a set, which is neither closed or open at the moment: $$A=\{(x,y,z)\in\mathbb{R}^3:0<x\leq 1, 0<y^2+z^2\leq x\}\subseteq\mathbb{R}^3$$ I know I can do different operations on the set to try to make it either an open or closed set, but I still do not know what to do exactly. I have not solved a problem of this kind before, so I would appreciate some advice.

Answer 1

That set is neither closed nor open. On the other hand, if $n\in\Bbb N$ and$$A_n=\left\{(x,y,z)\in\Bbb R^3\,\middle|\,0<x<1+\frac1n,0<y^2+z^2<x+\frac1n\right\},$$then each $A_n$ is open and $A=\bigcap_{n\in\Bbb N}A_n$. Therefore, $A$ is a Borel set.