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