$P$ is a probability measure and $A_1, A_2, ... \in F$ that is a sigma algebra.
$$P \bigg( \bigcup_{n=1}^{\infty} \bigcap_{k = n}^{\infty}A_k \bigg) = \lim_{n \rightarrow \infty}P \bigg( \bigcap_{k = n}^{\infty}A_k \bigg) $$
I am trying to prove this equality by using the definition of limit and finding for every $\epsilon > 0$ a $k \in N$ s.t.
$$ \Bigg| P \bigg( \bigcap_{k = n}^{\infty}A_k \bigg) - P \bigg( \bigcup_{n=1}^{\infty} \bigcap_{k = n}^{\infty}A_k \bigg) \Bigg| < \epsilon $$
for every $n \ge k$. But I am a bit stuck in finding such a $k$.
@Monolite - it is not an axiom per se, but it easily follows from the axioms of a measure.
Put $E_{n}:=\bigcap_{k=n}^{\infty}A_{k}.$ Notice that $E_{1}\subset E_{2}\subset E_{3}\subset\ldots$ (since if $j>l$, then $E_{j}$ is an intersection of fewer (of the same!) sets as $E_{l}$. Now put $E:=\bigcup_{n=1}^{\infty}E_{n}.$ Because the $\{E_{n}\}$ are nested, $E=\lim_{n\to\infty}E_{n}.$ Thus, our problem reduces to showing
$$(1)\;\;\;\;P\left(\lim_{n\to\infty}E_{n}\right)=\lim_{n\to\infty}P\left(E_{n}\right).$$
That is, $P:\mathcal{F}\to\mathbb{R}^{+}$ is "continuous" w.r.t. monotone sequences of sets (i.e. nested increasing [unions] and nested decreasing [intersections]).
Hint to prove (1)
Put $B_{n}:=A_{n}-A_{n-1}$ for $n=2,3,\ldots$ and $B_{1}:=A_{1}.$ Then $B_{j}\cap B_{l}=\emptyset$ for $j\neq l$, yet $E=\bigcup_{n=1}^{\infty}B_{n}$. Now apply the axiom of countable additivity.