 Solution

Suppose $u,v\in U_n$ both minimize the distance, with common value $d$. Apply the parallelogram identity to $x-u$ and $x-v$:The midpoint belongs to the linear subspace $U_n$, so its distance from $x$ is at least $d$. It follows that $\Vert u-v{\Vert }^2\le 0$, hence $u=v$. Thus the best approximation is unique.