Let xn be a submartingale such that supnnbspxnnbsplt infin

Question

Problem:

Let (X_n) be a submartingale such that sup_n X_n < ∞ a.s. and E sup_n (X_n-X_n-1) + < ∞. Show that X_n converges a.s.

Additional Information:

This problem is bascially from Mathematics and it is about submartingale converging.

Note: The solution is in handwritten format.