Suppose that a sequence is Cesaro summable. Prove.... Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that 
$$\lim_{n \to \infty }\frac{a_{n}}{n}=0$$

Suppose $(a_1 + \cdots +a_n)/n \to L \in \mathbb R.$ Then
$$\frac{a_n}{n} = \frac{a_1 + \cdots + a_n}{n} - \frac{n-1}{n}\frac{a_1 + \cdots + a_{n-1}}{n-1} \to L-L = 0.$$
Let's apply this to the Cesaro sums: Suppose $(S_1 + \cdots +S_n)/n \to L\in \mathbb R,$ where $S_n = a_1 + \cdots +a_n.$ By the above, $S_n/n \to 0.$ Applying the above again shows $a_n/n \to 0.$
(NOTE: This was edited from the original, where I only included the first result. Thanks to @Bungo for pointing this out.)
