- Choose a 'minimizing sequence' $\displaystyle d(x_n,y)\rightarrow \inf_{x\in A} d(x,y)$ and use the fact that $\displaystyle A$ is closed.
- Let there be two such points, $\displaystyle x$ and $\displaystyle z$. Show that any point $\displaystyle w $ on the line segment to join them satisfies $\displaystyle d(w,y)<\inf_{x\in A} d(x,y).$.