Let $\{f_n\}$ converge point-wise to $f$, where each $f_n:[a,b]\rightarrow \mathbb{R}$, and each $f_n$ is a continuous convex function. Furthermore, assume that $f$ is continuous. Prove that the convergence is uniform.
I was trying to do something like: Consider $a=a_0, where $a_{i+1}-a_i<\delta$, where the $\delta $ is such that $|x-y|<\delta$ imply that $|f(x)-f(y)|<\epsilon$. Then letting $N_i$ be such that $f_n(a_i)$ is $\epsilon$-close to $f(x_i)$, let $N$ be the max of $N_1,...,N_k$, and hence$$|f(x)-f_n(x)|\leq |f(x)-f(x_i)|+|f(x_i)-f_n(x_i)|+|f_n(x_i)-f_n(x)|$$I can make the first summand small since $f$ is continuos, and the second summand small by letting $n\geq N$. I am having trouble using the fact that they are convex.