How the sum of 2 random variables is also a random variable... Say I have two random variables $X$ and $Y$ defined on a probability space ($\Omega,\mathbb{F},\mathbb{P}$). 
To prove $X+Y$ is also a random variable, I need $\{\omega:(X+Y)(\omega)\le x\}\in \mathbb{F},\ \forall x\in \mathbb{R}.$ 
For positive random variables, I can simply write this set as an intersection of two sets, $\{\omega:X(\omega)\le x\}$ and $\{\omega:Y(\omega)\le x\}$, and invoke the property of the sigma algebra. 
Now, how do I extend this to arbitrary-valued random variables? Or is there an intersection that is irrespective of the value of the RVs? 
 A: For all $x \in \mathbb{R}$ you have
$$ \{\omega \in \Omega \::\: (X+Y)(\omega) < x \} 
=  \bigcup_{q \in \mathbb{Q}} \{\omega \in \Omega \::\: X(\omega) < x-q, Y(\omega) < q  \} $$
a countable union of sets of the form
$$\{\omega \in \Omega \::\: X(\omega) < x-q\} \cap \{\omega \in \Omega \::\: Y(\omega) < q\} \in \mathbb{F}$$
and we conclude $\{\omega \in \Omega \::\: (X+Y)(\omega) < x \} \in \mathbb{F}$
for all $x \in \mathbb{R}$, which proves $X+Y$ to be a random variable.
Please note, that in general
$$ \{\omega \in \Omega \::\: (X+Y)(\omega) \leq x \} \neq \{\omega \in \Omega \::\: X(\omega) \leq x\} \cap \{\omega \in \Omega \::\: Y(\omega) \leq x\}, $$
even if $X$ and $Y$ are assumed positive as there could be an $\omega \in \Omega$ with $X(\omega) = Y(\omega) = x >0$ and then $\omega$ would only be contained in the second one of these sets.

EDIT (suggested by Didier Piau):
Using the result above you also have
$$\{\omega \in \Omega \::\: (X+Y)(\omega) \leq x \} = \bigcap_{n \in \mathbb{N}} \{\omega \in \Omega \::\: (X+Y)(\omega) < x +\frac{1}{n} \} \in \mathbb{F}.$$
