EDB — 183

[183] Let $C\subset {ℝ}^n$ be a convex set; suppose that $f_i:C\to ℝ$ are convex, where $i\in I$ (a non-empty, and arbitrary, family of indices), and we define $f(x)=\underset{i\in I}{sup}{f}_i(x)$, where we suppose (for simplicity) that $f(x)<\infty$ for every $i$: show that $f$ is convex.