Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$.

Question

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$.

From here https://math.stackexchange.com/a/878635/140308 (proof attempt is there too)

Sorry for the confusion. How does that show that the series converges? What about n=1,...,m-1? Does our choice of A mean that the index now starts at M? Why/why not? I guess I got the idea right. Just don't know how to say it exactly