Can someone give a proof for the following statement, which is a generalization of the result in this question from domain $\mathbb{R}$ to $\mathbb{R}^m$:

Suppose $f_n: [-\infty, \infty]^m \rightarrow (-\infty, \infty]$ is a sequence of convex functions that converges pointwise to a non-constant convex function $f: [-\infty, \infty]^m \rightarrow (-\infty, \infty]$, then $\inf_{x \in\mathbb{R}^m} f_n(x)$ converges to $\inf_{x\in\mathbb{R}^m} f(x)$.