Intersection of an infinite sequence of open dense sets

Question

Let $\{G_n\}$ be a sequence of infinite open dense subsets of a complete metric space $X$. I have already proved intersection of $\{G_n\}$ is non-empty and now wants to prove this intersection is dense in $X$ (as it's claimed to be correct in a textbook), but got no idea on how to do so.

Answer 1

If $U$ is a non-empty open set then $U$ is itself a complete metric space under an equivalent metric. Consider $U\cap G_n$ in this space to see that $\cap G_n$ intersects $U$. Hence, $\cap G_n$ is dense.

Ref.: Open subsets of a complete metric space.