Equivalence of maps on l-infinity (involves limits, suprema and sums)

[TaPaKaH]

Homework Statement

Normed space $(l^\infty,\|\cdot\|_\infty)$ with subspace $S\subset l^\infty$ consisting of convergent sequences $x=(x_n)_{n\in\mathbb{N}}$.

Given a sequence of maps $A_n:l^\infty\to\mathbb{R}$ defined as $$A_n(x)=\sup_{i\in\mathbb{N}}\frac{1}{n}\sum_{j=0}^{n-1}x_{i+j}$$need to show that for any $x\in S$ one has$$\lim_{n\to\infty}A_n(x)=\lim_{n\to\infty}x_n.$$

Homework Equations

Already shown that for any $x\in l^\infty$ the sequence $A_n(x)$ is monotone decreasing in $n$ and is bounded by $\|x\|_\infty$ therefore is convergent.

The Attempt at a Solution

It's more or less clear to me that if a sequence converges then a sequence of "averages" also converges, but I am struggling to find a way to write this out explicitly.

 

[TaPaKaH]

nevermind, solved it myself via evaluations from both sides