Prove the theorem.
Let {Un} be {Fn}-adapted and integrable, and set
(i) {(Xn, Fn), n ≥ 0} is a martingale iff {(Un, Fn), n ≥ 0} is a martingale difference sequence, a submartingale iff {(Un, Fn), n ≥ 0} is a submartingale difference sequence, and a supermartingale iff {(Un, Fn), n ≥ 0} is a supermartingale difference sequence.
(ii) A martingale difference sequence has constant expectation 0; a submartingale difference sequence has non-negative expectations; a supermartingale difference sequence has non-positive expectations.