Is it possible anybody could supply a self-contained proof of the following reincarnation of Jensen's inequality? I couldn't manage to find a good one I liked anywhere$\ldots$
If $f$ is convex and $a_1, \ldots, a_n \in \mathbb{R}$, then$$f\left({1\over n} \sum_{i = 1}^n a_i\right) \le {1\over n} \sum_{i = 1}^n f(a_i).$$