Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 1 / 3D (Analysis I)

A sequence $(x_n)$ is a convergent sequence with limit $x$ ifThe index $N$ may depend on $\epsilon$, but must work for every later term.

Fix $\epsilon >0$. Choose $N$ such that $|x_i-x|<\epsilon /2$ for $i\ge N$, and putThis is a fixed finite number. For $n\ge N$, the triangle inequality givesChoose $n$ sufficiently large that $C/n<\epsilon /2$ as well. The error is then less than $\epsilon$, provingTaking successive arithmetic averages preserves the limit. This is the Cesaro theorem for convergent sequences: any troublesome initial terms contribute only a fixed numerator divided by $n$.