Prove that the limit of a sequence is 0. (Stolz-Cesaro theorem?)

Question

Prove that if $(a_{n+1} + a_n) \to 0$, then $\frac{a_n}{n} \to 0$. Is it possible to replace $0$ by some $g$?

It looks like using Stolz-Cesaro theorem is required here but I don't really have any ideas on how to start.

Answer 1

Let consider

$$\frac{b_n}{c_n}=\frac{(-1)^na_n}{n}$$

then by Stolz-Cesaro

$$\frac{b_{n+1}-b_n}{c_{n+1}-c_n}=(-1)^{n+1}a_{n+1}-(-1)^na_n=-(-1)^n(a_{n+1}+a_n)\to 0$$

and therefore

$$\frac{b_n}{c_n}=\frac{(-1)^na_n}{n}\to 0 \implies \frac{a_n}{n}\to 0$$