Let $$F(x) = \sum\_{a \leq x} f(a)$$ where $f: A \subseteq \mathbb{R} \rightarrow \mathbb{R} $ such that:

1. $f(a) \geq 0$ for any $a \in A$ and
2. $\displaystyle\sum\_{a \in A} f(a) = 1$

How do I prove that $$\lim\_{x\to -\infty} F(x) = 0\text{?}$$ I know how to do delta-epsilon proofs for limits that take $x$ to infinity, but I don't know any theorems that let me exploit the two conditions on the function allowing me to prove the following limit without having to "peak inside" the function to see what it looks like. My real analysis is very rusty.