Question: prove a sequence converges pointwise

has anybody any idea for this?  been stuck on it for a while now.

let f0 € V be any given function and define a sequence (f_n)_n€(No) of functions f_n € V by 

f₀ := f0 and f_n+1 =Af_n  for all n € (N₀)        #A could be the average of the four surrounding points to (i,j) or it could be an N x N matrix with spectral radius less than 1.  not entirely sure. I dont know what it should be but im sure one of you guys know.

prove that this sequence converges pointwise,

 i.e that for all i,j €  [N] x [N], f_oo(i,j) := lim_{n-> oo} f_n(i,j) exists.   and that Δf_oo=0 

it  says to be in the notation of (" http://www.mapleprimes.com/questions/201278-Fix-A-Syntax-Error-In-My-Simple-Function-please-Help")  but it doesnt matter if its not.  I can adapt to what its meant to be if I can get any way to prove it

Thanks in advance.