Let be a (discrete) non-negative sub-martingale, bounded in i.e. with for some fixed . Define by , and .
Then . Further, .
Proof: Fix and define the stopping time . Then which is -measurable and thus by The Stopping Time Lemma since
Also by Fubini
Using both of these facts and Fubini again
Hölder’s inequality states that for then . Now which is in hence . Thus
Hence from above
If then the inequality holds trivially.
and now apply monotone convergence to obtain