Convergence of Deterministic IFS

From the data we have assembled,

h(S0,S1) = 1/21

h(S1,S2) = 1/22

h(S2,S3) = 1/23

we see the general distance calculation is

h(Sn-1,Sn) = 1/2n

It follows that the sequence S0, S1, S2, ... is a Cauchy sequence in the Hausdorff metric.

To see this, take any e > 0 and take N large enough that

1/2N < e

Given any m, n > N (assume m > n),

h(Sn, Sm) ≤ h(Sn, Sn+1) + h(Sn+1, Sm) applying the triangle inequality ≤ h(Sn, Sn+1) + h(Sn+1, Sn+2)+ h(Sn+2, Sm) applying the triangle inequality again ≤ h(Sn, Sn+1) + h(Sn+1, Sn+2) + ... + h(Sm-1, Sm) applying the triangle inequality more times ≤ h(Sn, Sn+1) + h(Sn+1, Sn+2) + ... making this an infinite serties ≤ 1/2n+1 + 1/2n+2 + 1/2n+3 + ... applying the general distance calculation ≤ 1/2n < 1/2N< e

and the sequence is Cauchy, as desired.

With the Hausdorff metric, the set of compact subsets of the plane is complete, and so the sequence S0, S1, S2, ... converges to a limit.

Return to Hausdorff Distance Convergence Example.