How define the norm of $v+W$

Question

I'm trying to solve the following exercise:

Let $V$ be a normed linear space and let $W$ be a finite dimensional subspace of $V$. Show that, for all $v\in V$, there exists $w\in W$ such that $\|v+W\|=\|v+w\|$.

But I don't know how to deal with the $\|v+W\|$. It's well knowed that $$d(v,W)=\inf_{w\in W}\|v-w\|.$$ Is that ok define $$\|v+W\|=\inf_{w\in W}\|v+w\|,$$ or I'm wrong? Maybe use a $\sup$?

Answer 1

Yes, the usual definition of $\lVert v+W\rVert$ is $\inf\{\|v+w\|\mid w\in W\}$. So, the problem consists in proving that this infimum is actually a minimum, under the assumption that $W$ is finite dimensional (the assertion is false in general).