In this appendix we prove the lemma required by the proof of Theorem 3.3:

where , there exists a bound such that with probability one, and for all . Assume that for all , uniformly in w.p.1 and w.p.1. Then w.p.1.

Fix arbitrarily. Furthermore fix a sequence of numbers ( ) to be chosen later.

Let
and
. Then

Thus, we have that . Now define . Since , . Consequently, if , then holds for all , as well. From now on, we will assume that .

Let . Since , the process

holds with probability . Now let us restrict ourselves to those events for which inequality (16) holds. Then we see that the process

bounds from above from the index . The process converges to w.p.1 uniformly over , so the above argument can be repeated to obtain an index such that (16) holds for with probability .

Since , and . So there exists an index for which . Then inequality (15) can be satisfied by setting so that holds and letting .