proof of Jensen’s inequality

We prove an equivalent, more convenient formulation: Let $X$ be some random variable, and let $f(x)$ be a convex function (defined at least on a segment containing the range of $X$). Then the expected value of $f(X)$ is at least the value of $f$ at the mean of $X$:

$$𝔼[f(X)]\ge f(𝔼[X]).$$

Indeed, let $c=𝔼[X]$. Since $f(x)$ is convex, there exists a supporting line for $f(x)$ at $c$:

$$\phi (x)=\alpha (x-c)+f(c)$$

for some $\alpha$, and $\phi (x)\le f(x)$. Then

$$𝔼[f(X)]\ge 𝔼[\phi (X)]=𝔼[\alpha (X-c)+f(c)]=f(c)$$

as claimed.