I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that
$$E\left(X_n \right) \to c$$
as well. This is quite intuitive as all probability mass is bounded away from infinity in this case, but I am having a hard time formalizing this notion. Could you please give me some hints? Please keep it simple, though. Thank you.