Realization vs Random Variables in Formulae
Solution 1:
Random variables are, by definition, functions from a probability space $(\Omega, {\mathcal F}, \mathbb{P})$ to (usually) the real numbers. This fact is often concealed in elementary presentations of probability theory.
Thus, adding random variables is just adding functions. Of course you then have to understand what you mean by the left hand side above (a sequence of functions) converging to $0$.