Complete embeddings into $Fn(\kappa \times \omega, \omega)$

Question

I have problems to show the following:
Let $\kappa$ be a regular uncountable cardinal, $\lambda < \kappa$ and let $\mathbb{P}=Fn(\kappa \times \omega, \omega)$, $\mathbb{Q}=Fn(\lambda \times \omega, \omega)$ and $\mathbb{R}=Fn(J,\omega)$. $J= \{\alpha : \lambda \leq \alpha < \kappa \}$. Show that each of the posets $\mathbb{Q}$ and $\mathbb{R}$ completely embeds into $\mathbb{P}$.
My problem is that i don't know how to show that.
The case for $\mathbb{Q}$ seems trivial since $\lambda < \kappa$. But is it really that trivial?
I have no idea how to show that $\mathbb{R}$ completely embeds into $\mathbb{P} $.

Definition
For an infinite index set $I$ we defined $Fn(I,\omega)$ as the partial order of all finite partial functions $p:I \rightarrow \omega$ with extension relational superset.