I have a question that I encountered during my internship:
Consider a convergent sequence of continuous, convex functions $\{f_n(x)\}_n$ defined in $\mathbb{R}^M$. These functions are uniformly Lipschitz continuous, that is, $\exists C\in\mathbb{R}$ such that:
$\forall x,y \in \mathbb{R^M},\forall n\ge1\quad |f_n(x)-f_n(y)|\le C|x-y|.$
Furthermore, each function $f_n(x)$ has a minimizer. The properties of simple convergence and uniform Lipschitz continuity allow us to prove that the convergence is uniform in any compact of $\Bbb R^M$.
My question is:
Can we demonstrate that $\inf_{\Bbb R^M}f_n(x)$ converges to $\inf_{\Bbb R^M}f_{\infty}(x)$ as $n\rightarrow\infty$, where $f_{\infty}(x)$ is the limit of $f_n(x)$ and it is supposed that $\inf_{\mathbb{R^M}}f(x)$ is finite?
Thanks a lot!