Let $<A_{i}>$ be a countably infinite partition of sample space $\Omega$. Then the probability of an event $A$ is given by,
$$P(A) = \sum_{n = 1}^{\infty} P(A_{i})\cdot P(A|A_{i})$$
The derivation of the theorem uses the countable additivity axiom.
How are we so sure that the sum converges?
We are sure exactly because of countable additivity. We can rewrite the result $$ P(A) = \sum_i P(A\cap A_i).$$ This follows immediately from countable additivity because the sets $A\cap A_i$ are disjoint and $\bigcup_i (A\cap A_i) = A.$ (And these two facts follow from the fact that $A_i$ is a partition of $\Omega$.)
I'm not sure why the convergence of the sum would be a separate consideration. In fact if we were working in an infinite measure where $P(A)=\infty,$ it would still be true and the sum wouldn't converge. But since we're in a probability space, $P(A)$ is finite.