Let S= { 0,1,2,3,4,5,6,7,8,9} and let a(1),a(2) ... be an infinite sequence of elements of S such that a(i+1) is determined by a(i) for every i. That is a(i+1) = f(a(i)) for some function f from S to S.

Show that there exists a positive integer k such that a(2k) = a(k).

This is my attempt

There is no a(0) .. so a(1)=f(a(0)) , a(2)=f(a(1)) a(3)=f(a(2)) .. ......

??

Now im just confused.