Hello,

This is Cesaro's mean. And you want to prove Cesaro's lemma.

There is a proof in the French Wikipedia (Lemme de Cesàro - Wikipédia), but not in the English one.

I'll translate and adapt it here...

can be translated this way :

Let

Let's assume n>N :

By the triangle inequality, we have :

(because N/n (epsilon/2) >0)

But doesn't depend on n. So

This means that for any , there exists an integer N' such that for all n>N', we have :

By combining this latter inequality to the previous one, we have, for :