I have some problems proving the following:
Let $f: \mathbb{R}^n \rightarrow ( - \infty, \infty]$, $f_i: \mathbb{R}^n \rightarrow ( - \infty, \infty], \, i \in I$ be convex functions. Prove that $f = sup_{i \in I} f_i$ if and only if $ \text{epi} f = \bigcap_{i \in I} \text{epi} f_i$.
I managed to show the first direction. But i have troubles showing the back direction (see image). I don't know how to get from the epi to the definition of f.
I would be grateful about some help!
Here's what i got so far
