Given a sequence of non-negative random variables $(X\_i)\_{i\in\mathbb{N}}$, I would like to show that

$$ \mathbb{E}[\sum\_{i=1}^\infty {X\_i}] < \infty$$

implies that 

$$ \sum\_{i=1}^\infty {X\_i} < \infty $$

almost surely

The approach that I have in mind is to condition the expectation on the value of $\sum\_{i=1}^\infty {X\_i}$ as follows:

$\begin{align}
\mathbb{E}[\sum\_{i=1}^\infty {X\_i}] =& \mathbb{E}[\sum\_{i=1}^\infty {X\_i} \ \Bigg | \sum\_{i=1}^\infty {X\_i} < \infty] \mathbb{P}[\sum\_{i=1}^\infty {X\_i} < \infty] + \mathbb{E}[\sum\_{i=1}^\infty {X\_i} \ \Bigg | \sum\_{i=1}^\infty {X\_i} = \infty] \mathbb{P}[\sum\_{i=1}^\infty {X\_i} = \infty] \end{align}$

and then say that since 

$\mathbb{E}[\sum\_{i=1}^\infty {X\_i}] < \infty$

I can then argue that the above conditions imply that 

$\mathbb{E}[\sum\_{i=1}^\infty {X\_i} \ \Bigg | \sum\_{i=1}^\infty {X\_i} < \infty] \mathbb{P}[\sum\_{i=1}^\infty {X\_i} < \infty] < \infty$

$\mathbb{E}[\sum\_{i=1}^\infty {X\_i} \ \Bigg | \sum\_{i=1}^\infty {X\_i} = \infty] \mathbb{P}[\sum\_{i=1}^\infty {X\_i} = \infty] < \infty$

Given that $\mathbb{E}[\sum\_{i=1}^\infty {X\_i} \ \Bigg | \sum\_{i=1}^\infty {X\_i} = \infty] = \infty $, this must mean that $\mathbb{P}[\sum\_{i=1}^\infty {X\_i} = \infty] = 0$ 

Although this explanation makes sense intuitively, it doesn't seem formal enough (in particular it relies on the notion that $0 \times \infty = 0$). Is there a more elegant or formal approach?