Prove that $\cap A_i$ is dense

Question

Let $M$ be a complete metric space and let $\{A_i\}$ be a countable collection of dense open subsets of $M$. Prove that $\cap A_i$ is dense.

This prblem is an exercise (6.3.6) from the book Set Theory and Metric Spaces by Irving Kaplansky.

Since the space $M$ is complete, every Cauchy sequence is convergent to a point (in $M$). I would like to show that every point in $M$ is either in $\cap A_i$ or is a limit point of $\cap A_i$. However, I have no idea where to start.

Answer 1

This problem is one of the equivalent form of Baire Category Theorem