How to prove that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$

Question

Let $a_1,a_2,...$ be nonnegative numbers whose sum is $1$ and let $x_1,x_2,...>0$. I want to show that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$.

This looks awfully similar to the $AM \geq GM$ inequality. In fact, the AM-GM inequality is a special case of this.

I was thinking of using Jensen's inequality which states, if a function $\phi$ is convex on all of $\mathbb{R}$ and $f$ is Lebesgue integrable on $[0,1]$ and $\phi \circ f$ is nonnegative on $[0,1]$, then $\phi(\int_0 ^1 f)\leq \int_0 ^1 \phi \circ f$.

Answer 1

Let $\mu$ be the measure on the power set of $\mathbb N$ with $\mu \{n\}=a_n$. Let $\phi (x) =-\log x$. $\phi$ is convex. Apply Jensen's inequality to $f(n)=x_n$.

Answer 2

In case the $a_i$ are rational, this is the AM-GM inequality (use a common denominator). But also both sides are continuous, so you get the real case in the limit.