Let $(f_{i})_{i\geq 1}$ be a sequence of concave functions which converges uniformly to a function $f$. Then $f$ is a concave function and the sequence $(f_{i}^{\vee})_{i\geq 1}$ converges uniformly to $f^{\vee}$. In particular, there is some $i_{0}\geq 1$ such that ${\operatorname{dom}}(f_{i})={\operatorname{dom}}(f)$ and $\operatorname{stab}(f_{i})=\operatorname{stab}(f)$ for all $i\geq i_{0}$.