BALTIC-WAY 2017 Problem 1

Let $a_{0}, a_{1}, a_{2}, \ldots$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2} \geq a_{n}$ for all positive integers $n$. Show that

$$\frac{a_{0}+a_{n+1}}{2} \geq \frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$$

holds for all positive integers $n$.