Existence of an element from a quotient space

Question

Let $V$ be a finite-dimensional complex inner product space and $U\subset V$ , a subspace of $V$ and $v \in V $, then it exists a unambiguous $w \in v + U$, such that $$\Vert w\Vert= min\{\Vert v'\Vert : v' \in v+U\}$$ While I know that this $w$ is (probably?) induced by the quotient space $V/U$ ,I have trouble understanding how to proceed.