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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.

  • 1
    How do you extend this result to $\mathbb{R}^n$?2014-04-30
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    Nice result, I was not aware of it. Do you have references for it? Did you discover it yourself or did you learn it somewhere?2015-12-03
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    @DelioMugnolo I saw it in a test a long time ago.2016-02-07

3 Answers 3