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

[TaPaKaH]

Homework Statement

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

Given a sequence of maps [itex]A_n:l^\infty\to\mathbb{R}[/itex] 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 [itex]x\in S[/itex] one has$$\lim_{n\to\infty}A_n(x)=\lim_{n\to\infty}x_n.$$

Homework Equations

Already shown that for any [itex]x\in l^\infty[/itex] the sequence [itex]A_n(x)[/itex] is monotone decreasing in [itex]n[/itex] and is bounded by [itex]\|x\|_\infty[/itex] 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