Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each $f_n$ is continuous and concave, then the convergence is uniform.
Main question: is the above true and, if so, could you please provide a proof (or a reference to a proof)? I know that if $f_n$ is continuous, then it is uniformly continuous but I can't proceed further.
Edit 2: as Nate's answer below demonstrates, the claim is false given the premises as they are. What if we further assume that $f$ is continuous?
Sub-question if you have time: can you please give a few references concerning sufficient conditions that allow the implication from pointwise convergence to uniform convergence? I know of Dini's Theorem and Arzelà-Ascoli Theorem.
Thank you!
Edit 1 to add context: on page 390 of (Monfort and Gourieroux (1995, Vol II)), it is stated:
a family $Q_n(\omega,\cdot)$ of concave functions converging pointwise to a function $Q_\infty(\cdot)$, which is necessarily concave, converges uniformly to that function over any compact subset
Here, $\omega$ can be taken as given and $\cdot$ is a place holder for $\theta\in\Theta$ with $\Theta\subset\mathbb{R}^p$ is compact and convex. Hence, the question above.
This is not true. Take $X = [0,1] \subset \mathbb{R}^1$ and $f_n(x) = 1-x^n$.