Proof thr following of cauchy's theorems:
If (a$_m)m\geq$0 is a convergent sequence in $\mathbb{R}$ and $b=\lim_{m\to \infty} a_m$ then
$$\lim_{m\to \infty} \frac{a_0 + a_1 + ... + a_m}{m+1}= b $$
NOTE: We have only recently started learning about series and sequences and can therefor only use very limited definitions and theorems such as the definition of a sequence and subsequence,definition of convergent sequences,... and little more. So the most basic proof would be usefull.