Consider the norm $\|(x,y)\|=|x|+|y|$ on $\mathbb{R^2}$. If $K$ is any non-empty closed subset of $\mathbb R^2$, I want to show that if $x\in \mathbb R^2$, then there exists $x_0\in K$ such that $\|x-x_0\|=d(x,K)$.
Since $\mathbb R^2$ is not compact we can't say that $K$ is compact. Then How to show?
I have taken the problem from Conway's "A Course in Abstract Analysis".