Convergence of Minima to Minima for sequences of Convex Functions

Question

Let $V$ be a vector space with a metric and let $\varphi_{n}:V \to \mathbb{R}$ ($n\in\mathbb{N}$) be a sequence of convex functions converging pointwise to $\varphi$. If for every $n$, $v_{n}\in V$ satisfies $$v_{n}\in \arg\min_{v\in V}\varphi_{n}(v) $$ and $v_{n}\to_{n} v$, is it true in general that $$v\in \arg\min_{v\in V} \varphi(v)?$$ If not, does the convergence hold if $V$ is normed and $\varphi_{n}$ is assumed continuous?