I am not able to understand this proof of the uncountability of dense $G\_\delta$ sets in $\Bbb R$.

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> If $(G\_n)$ is a sequence of open dense sets in $\Bbb R$ and if $\bigcap\_{n=1}^\infty G\_n = \{x\_1,x\_2,...\}$ then the sets $\tilde G\_n = G\_n\setminus \{x\_n\}$ are still open and dense, but $\bigcap\_{n=1}^\infty \tilde G\_n = \emptyset$, contrary to Baire's theorem. Thus, $\bigcap\_{n=1}^\infty G\_n$ is uncountable.
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1. What is a dense $G\_\delta$ subset of $\Bbb R$? I think it should be
$$G\_\delta = \bigcap\_{n=1}^\infty G\_n, G\_n \text{ open}$$ such that $$\overline{\bigcap\_{n=1}^\infty G\_n} = \Bbb R$$
Then why does the author insist that $G\_n$ be dense too? Isn't it enough for $G\_n$ to be open by definition of $G\_\delta$?
2. Why $\tilde G\_n$ are open and dense? I see that $\tilde G\_n$ are open because deleting a point from an open set preserves its openness, but what about density?
3. Why is $\bigcap\_{n=1}^\infty \tilde G\_n = \emptyset$? If this is true, then I see the contradiction. Need help showing that $\bigcap\_{n=1}^\infty \tilde G\_n = \emptyset$ though.

Merci!