Embedding of linear order into $\mathcal{P}(\omega)/\mathrm{fin}$ I try to prove following problem (in Kunen):

Assume $\mathrm{MA}(\kappa)$ and $(X,<)$ be a total order with $|X|\le\kappa$, then there are $a_x\subset \omega$ such that if $x<y$ then $x\subset^* y$.
  ($x\subset^* y$ if $|x-y|<\omega$ and $|y-x|=\omega$.)
Hint. Let $P$ be a set of pairs $(p,n)$, where $p$ is a partial function from $X$ to $\mathcal{P}(n)$ and the domain of $p$ is finite. 
  Define $(p,n)\le (q,m)$ iff $\operatorname{dom}p\supseteq \operatorname{dom}q$, $n\ge m$, $p(x)\cap m=q(x)$ for each $x\in \operatorname{dom} q$ and
  $$\forall x,y\in \operatorname{dom}q:x<y\implies p(x)-p(y)\subseteq m.$$
  Use the $\Delta$-system lemma to prove that $(P,\le)$ has the c.c.c.

I prove $P$ has the c.c.c. and if $G$ is a filter which intersects the dense sets
$$D_x=\{(p,n):x\in\operatorname{dom}p\}$$
 over $P$ then $a_x:= \bigcup\{p(x)\mid\exists n<\omega: (p,n)\in G\text{ and }x\in\operatorname{dom}p\}$ satisfies $|a_x-a_y|<\omega$ if $x<y$. But I don't know how to prove $|a_y-a_x|=\omega$. I think some dense sets are added but which sets are added? Thanks for any help.

For $x < y \in X$ and $N \in \omega$, consider the dense set $D_{x, y, N} = \{(p, n) \in P : x, y \in \text{dom}(p) \text{ and } (\exists m > n) (m \in p(x) - p(y))\}$.
