$\inf\limits_{\{(x,y)\in\mathbb{R}^2:x^2+y^2\le 1\}} \sqrt{x^2+y^2}=\min\limits_{\{(x,y)\in\mathbb{R}^2:x^2+y^2\le 1\}} \sqrt{x^2+y^2}=\sqrt{x^2+y^2}$$$$=\sqrt{\min\limits_{\{(x,y)\in\mathbb{R}^2:x^2+y^2\le 1\}} (x^2+y^2)}=\sqrt0=0$ (since $x^2+y^2\ge 0$).
I was wondering if there is more to this argument to make it more rigorous or elegant?
In the polar coordinate system we have $\inf\limits_{r\in\mathbb{R}:0\leq r\le 1} r=\min\limits_{r\in\mathbb{R}:0 \leq r\le 1} r=0$