Let $f_n\colon \mathbb{R}^3\to\mathbb{R}$ be continuous convex non-decreasing functions for all $n\in\mathbb{N}$ and $f_n\to f$ pointwise. I want to prove that the convergence is uniform in $\mathbb{R}^3$.
At first I wanted to use Dini's theorem but $\mathbb{R}^3$ is not compact. I understand that convexity is the key property here. Are there any other well known theorems that use convexity, in order to get rid of compactness?
The result is not true. Consider the following example in one dimension, that can be easily adapted to 3-dimensional space: $$ f_n(x)=\frac1n\,e^x. $$ Each $f_n$ is increasing, convex and continuous. $f_n$ converges pointwise but not uniformly to $0$.