Assume that, given iid taking values in and some extra random variable , you build a decision tree with cells . Note that are deterministically obtained from and but we do not stress the dependence on these variables in the notation.

Assume that where are iid, centered with finite variance and the are independent from the and . In what follows . Denote where is the cell containing and . Define the diameter of any cell by

The proof is based on the following lemma:

Proof of the Lemma: Write Thus, and since is arbitrary, this concludes the proof.

Denote and since is continuous on the compact set , it is bounded and uniformly continuous. Therefore, is a bounded function that converges to as . Then, denoting , we have by the triangular inequality, The proof follows by applying the lemma to the random variables on with the function that is bounded on and continuous at and then to the random variables taking values on with the bounded function on which is continuous at .