I have problem with the following question

Let $\displaystyle \xi_1, \xi_2,...$ be independent and identically distributed random variables with $\displaystyle E|\xi_i|< \infty $. Show that

$\displaystyle E\left(\xi_1|S_n,S_{n+1},...\right)=\frac{S_n}{n} \ \ \ (a.s)$

where $\displaystyle S_n= \xi_1+...+\xi_n$

I proceed this way:

I want to know wether this is true or not.

$\displaystyle S_n = E(S_n|S_n)=E\left(S_n|S_{n},S_{n+1},S_{n+2},...\ri ght)$ ???

Now since $\displaystyle \xi_i$ are independent and identically distributed, then

$\displaystyle E(\xi_1)=E(\xi_2)=...=E(\xi_n) $

$\displaystyle E \left(S_n|S_{n},S_{n+1},S_{n+2},...\right)=n.E\lef t(\xi_1|S_{n},S_{n+1},S_{n+2},...\right)$

so finally

$\displaystyle E\left(\xi_1|S_n,S_{n+1},...\right)=\frac{S_n}{n} \ \ \ (a.s)$