Given a sequence $(a_n)$ that is non-increasing (i.e $a_{n+1}\leqslant a_n$);
We construct the following auxiliary sequence:
$$ b_n = \frac{a_1+a_2+\dots+a_n}{n}$$
I need to prove that:
$$ \displaystyle{\lim_{n\rightarrow \infty}} a_n = 0 \Rightarrow \lim_{n\rightarrow \infty} b_n = 0$$
All I was able to prove is that $\lim b_n \leqslant a_1$ and that $\lim b_n \geqslant 0$
