Suppose that is a sequence in . Define a sequence by , . Prove that if converges to then converges to .
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 :