Given $f:\mathbb{R}\to\mathbb{R}$ is convex, then for any $x_1,\dots ,x_n$ and $\lambda_1,\dots ,\lambda_n$ with $\sum_{i=1}^n\lambda_i = 1$,$$ f\left(\sum_{i=1}^n\lambda_i x_i\right) \leq \sum_{i=1}^n\lambda_i f (x_i) . $$ Use this equation to prove E[f(X)] $\geqslant$ f(E[X]) for any random variable X that takes on only finitely many values.
My thought is that it's similar to Jensen's inequality, but I don't see how it specifically works for this given condition.
Thank you so much for your help!