Convergence of Deterministic IFS

To see A = T (A), note that because the Tk(B) are nested, for each M

T (∩k=1M T k(B)) = T(T M(B)) = T M+1(B) = ∩k=1M+1 T k(B).

Taking the M → ∞ limit, and using the continuity of T to interchange T amd the limit, we obtain

T (A) = A

Return to the proof of the theorem.