From the book "Topology without tears".
The proof of (i) is trivial. But in (ii) the sentence "Let $\{X_n\}$ be a sequence of open dense subsets of $Y$". It looks to me that it should be $X$ instead of $Y$. Otherwise it is a tautology: if $X_n$ is an open dense subset of $Y$ then $\{X_n \cap Y\} = \{X_n\} \in Y$ - by assumption of the problem and we don't need to prove anything?