The book says that statement 2 is a direct consequence of statement 1. I don't see how they prove statement 2 directly from statement 1, can you please help me?
Statement 1:
A complete metric space $(\Omega,\rho)$ is not the union of a countable collection of nowhere dense sets.
Statement 2:
Let $\{G_n\}_{n=1}^\infty$ be a sequence of dense open subsets of a complete metric space. Then $\cap_{n=1}^\infty G_n$ is dense.
So I struggle to see $1\rightarrow 2$. What I do know is that since each $G_n$ is dense, then $G_n^c$ is nowhere dense.(A set is nowhere dense iff the complement og its closure is dense, an $G_n^c$ is already closed, since $G_n$ is open). Hence I do know that $\cap_{n=1}^\infty G_n$ is nonempty, since it's complement is a countable union of nowhere dense sets, and from statement 1, this can't be the whole set.
But I only get that it has atleast 1 element, that is a long way from proving that it is dense.