Hello,

Just a quick thought.

Write the definition of a.s. convergence :

Let . We want to show, through the MCT, that for any .

The aim is to prove that by contradiction (I think that's the way...). So let's suppose .

means that .

Assuming that (it's true for any epsilon and we usually choose a small epsilon), it means that , since Xn takes only integer values.

And then try to finish by considering the probability (*) and the fact that . It should be possible to finish the proof (but I've merely written what I was thinking, so I'm not sure...)