dense subsets in metric spaces I am stuck on this problem. Any help? 
Problem
a)- Let $X$ be a complete metric space and let $V_{n}$, $n=1,2,3,...$ be open and dense sets. Prove that $\bigcap_{n=1}^{\infty }V_{n}$ is dense in $X$ .
b)- Use part a) to prove that the set of irrational numbers cannot be written as a union of countably many closed subsets of $R$.
 A: Part a) is a famous theorem, the Baire Category Theorem. 
b) Suppose  $\mathbb  R -\mathbb Q =\bigcup_{n\in \mathbb N} F_n $,where $F_n$ is closed for every $n$.so we can write $\mathbb Q$ as $\bigcap_{n\in \mathbb N}(\mathbb R -F_n)$ .
Note that every $\mathbb R -F_n$ is open and contains $\mathbb Q$, therefore dense.
Let $\mathbb Q = \{q_1,q_2...\}=\bigcup_{n\in \mathbb N} \{q_n\}$
then $\emptyset=\mathbb Q- \mathbb Q =  \bigcap_{n\in \mathbb N}(\mathbb R -F_n-\{q_n\} )$.
Note that the set on the right side of the equality is intersection of countable many open dense sets. By a) it must be dense.
Contradiction.
A: Sure.  Part a) is not a "problem" but a famous theorem, the Baire Category Theorem.  Note that the linked to wikipedia article gives a proof.  
(By giving this answer, I am acting on my opinion that it is not reasonable to expect a student to come up with a proof of this on her own, or at least that a reasonable response to being asked to do so is to look up the proof.)
Hint for part b): take complements.
