## Proving convergence of expected value

Hello,

I have been lately thinking of how to proof the following:

Let $X: \Omega \rightarrow \Re$ be a random variable and $X \geq 0$ then $\sum_{i=1}^\infty x_i P(X = x)$ converges to the expected value of $X: \Omega \rightarrow \Re$. The only thing that I am given is that $E[X] := \int_\Omega X(\omega)d P(\omega)$ provided that $\int_\Omega |X(\omega)| dP(\omega) < \infty$. Moreover, X is Borel measurable.

As $X \geq 0$ then $X_n \leq X_{n+1}$ (trivial to show) for all $n \in \mathbb{N}$ where $X_n = \sum_{i=1}^n x_i$. After going through all sorts of stuff, I have a feeling that this relevant, but I cannot prove that $\lim_{n \rightarrow \infty} X_n = X$.