Prove that set is not a compact

Question

I need to prove that in metric space $\Bbb{R}^2$ the set $$1\lt x^2+y^2 \le 4 $$is not compact.

I know theorem, that $$A\subset\Bbb{R}^n \; is \;a \; compact \iff A \; is \; bounded \; and \;closed.$$ But this doesn't work for me, because I need to prove it by definition of compact.

Answer 1

The family $B_n=\{(x,y) | 1+\frac{1}{n} <x^2+y^2 \leq 4\}$ is an open cover with no finite subcover.