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...)