Define $\mathbb{P}$ = {$\left<s,F\right>:s\in\omega^{<\omega}\wedge F\subseteq\omega^\omega\wedge|F|<\aleph_0$}. Define $\left<t,G\right>\leq\left<s,F\right>$ if and only if
(1) $s\subseteq t$ and $F\subseteq G$, and
(2) $\forall n\in\text{dom}(t)\backslash\text{dom}(s)\forall f\in F[t(n)>f(n)]$.Prove the following:
(a) $\left<\mathbb{P},\leq\right>$ is c.c.c.
(b) If $\text{MA}_\kappa$ holds, then for every $\mathscr{F}\subseteq\omega^\omega$ with $|\mathscr{F}|\leq\kappa$, there exists $h\in\omega^\omega$ such that for each $f\in\mathscr{F}$, |{$n\in\omega:h(n)\leq f(n)$}|$<\aleph_0.$
I know how to prove $\left<\mathbb{P},\leq\right>$ is c.c.c, but for (b), when MA$_\kappa$ holds, that is for any c.c.c. $\left<\mathbb{P},\leq\right>$ and a family $\mathscr{D}$ of $\leq\kappa$ dense subsets of $\mathbb{P}$, then there is a filter $G$ in $\mathbb{P}$ such that $\forall D\in\mathscr{D}(G\cap D\neq0)$, I do not really see the direction to show the second statement. Any hints will be appreciated, thank you.
Hint. For each $f\in\mathscr{F}$, define $D_f$ by $$D_f = \{(s,F) \mid f\in F\}.$$ Then you can see that $D_f$ is dense. Then examine the first coordinate of elements of an $\mathscr{F}$-generic filter.