Adding one Random Real I'm studying Kanamori's book The Higher Infinite and now I'm stuck.
I want to prove that forcing with Borel sets of positive measure adds one random real.
Let me state the theorem:
Let $M$ be a countable transitive model for ZFC. 
$\mathcal{B} = \{A \in Bor(\omega^\omega): \mu(A) > 0\}$ and $p \leq q \leftrightarrow p \subset q.$
Let $G$ be $\mathcal{B}^{M}$-generic over $M$. Then there is a unique $x \in \omega^\omega$ such that for any closed Borel code $c \in M$,
$$x \in A_{c}^{M[G]} \leftrightarrow A_{c}^{M} \in G$$
and $M[x] = M[G]$.
The proof first asserts:
"(*)For any $n \in \omega$,
$$\{C \in \mathcal{B}^{M}: C\text{ is closed }\land \exists k (C \subset \{f \in \omega^\omega: f(n) = k\})\},$$
is dense in $\mathcal{B}^M$."
My first question: Why? Is it obvious? I think I need at least a hint!
[...]Some irrelevant comments.
"Arguing in $M[G]$: there is a unique $x \in \omega^\omega$ specified by
$$\{x\} = \bigcap \{A_{c}^{M[G]}: c \in M \text{ is a closed code }\land A_{c}^{M} \in G\}$$
since this is an intersection of closed sets with the finite intersection property and (*) holds"
Can someone explain it to me more detailed?
Thank you!

Let $A_k=\{f\in\omega^\omega: f(n)=k\}$. Fix $C\in \mathcal B^M,$ note that  $\bigcup_{k\in\omega}A_k=\omega^\omega$ thus $C=C\cap (\bigcup_{k\in\omega}A_k)=\bigcup_{k\in\omega}(C\cap A_k)$. It follows that there is a $k$ so that $C\cap A_k$ has positive measure. Hence $C\cap A_k\leq C$ and satisfies $(*)$. 
