I have a question during my intership. Given a convergent sequence of continuous et convex functions $\{f\_n(x)\}$ defined in $\mathbb{R}^M$. These functions are uniformly Lipschitz continuous which means that there exist a constant C such that


$$|f\_n(x)-f\_n(y)| \leq C|x-y|, \quad \forall x,y \ \textrm{ in } \ \mathbb{R}^M \ \textrm{ and } \ n \geq 1 . $$


Furthermore, each function $f\_n(x)$ has a minimizer.


So the simple convergence + uniformly Lipschitz continuous allow us to prove the convergence is uniform in any compact of $\mathbb{R}^M$.


Now my questionn is that whether we can demonstrate


$$\inf\_{x \in \mathbb{R}^M} f\_n(x) \to \inf\_{x \in \mathbb{R}^M} f(x)$$
whenever $n$ goes to $\infty$?


Here $f(x)$ is the limit of $f\_n(x)$ and is supposed that $\inf\_{x \in \mathbb{R}^M} f(x)$ is finite.


Thanks a lot!



convergence of infimum
I think there are counterexamples: consider $f\_n(x)=\arctan(\frac{x}{n})$, then $f\_n(x)$ converge to $0$ and are uniformly Lipschitz continuous since there derivatives are smaller than $1$. However the inf of $\arctan(\frac{x}{n})$ is $-\frac{\pi}{2}$，which is not $0$.


