EDB — 183

[183] Let \(C⊂ ℝ^ n\) be a convex set; suppose that \(f_ i:C →ℝ\) are convex, where \(i∈ I\) (a non-empty, and arbitrary, family of indices), and we define \(f(x)=\sup _{i∈ I} f_ i(x)\), where we suppose (for simplicity) that \(f(x){\lt}∞\) for every \(i\): show that \(f\) is convex.