Infinite set is dense set in cofinite topology

Question

Define $$\tau = \{U \subset X; U^c \ \text{is finite or} \ U=\emptyset\}.$$ Let $X$ be an infinite set and $Y \subset X$ be an infinite subset of $X$. Prove that $Y$ is dense in the cofinite topology and that $(X,\tau)$ is separable.

My attempt:

Let $x \in X$ and $O$ be an open set that contains $x$. Using the definition we have that $X\setminus O$ is finite. If $Y \cap X=\emptyset$, then $Y \subset X\setminus O$ which is absurd. So $Y$ intersects $O$ and, as $x$ is a neighbourhood of $O$, $x \in \overline{Y}$ and we have $\overline {Y}=X$.

Is this enought or we have some mistake?

For the second part, i don't know to prove that $(X,\tau)$ is separable, any hint?

Answer 1

1. It is enough (I would change the phrasing "is absurd" - but I am not a native english speaker)

2. "For the second part, i don't know to prove that $(X,\tau)$ is separable", you just did prove it - think about the definition of separable space.