Convergence in vector space

I have two questions. Let $V$ be a subspace of finite-dimensional vector space $U$. Suppose $u_kto u_0$ in $U$ as $ktoinfty$.

First, for any $fin U’$, is it true that $f(u_k)to f(u_0)$? If so, why is this true?

Second, does $u_kto u_0$ in $U$ imply that $[u_k]to[u_0]$ in $U/V$ as $ktoinfty$ where the norm of $U/V$ is induced by the norm of $U$ by $vertvert[u]vertvert=inf{vertvert vvertvert :vin [u]}$ where $[u]in U/V$ and $uin U$?

Note: $U’$ denotes the dual space of $U$ (space of linear functionals)