Originally Posted by **CaptainBlack**

Sketch of a proof (you may have to fill in detail):

1. There exist a $\displaystyle \delta >0$ such that for all $\displaystyle v\epsilon V$ and $\displaystyle w\epsilon W$ $\displaystyle d(v,w)>\delta$.

For if this were not the case we could construct a sequence $\displaystyle v_i \epsilon V, i=1,2,3..$ with

limit $\displaystyle w \epsilon W$, but this contradicts $\displaystyle V$ closed; as closed means the limit must be in $\displaystyle V$.

(A closed set contains all its limit points)