Suppose $M$ satisfies the $CH$ and that we force over $M$ with $\mathbb{P}=Fn(I,2)$ where $(\omega_{2} \leq |I|))^{M}$, that is, with the finite-support functions from $I$ to $2$. If $f \in M[G] \cap \omega^{\omega}$, then is there necessarily a function $g \in M \cap \omega^{\omega}$ for which {$n:f(n) \leq g(n)$} is infinite? Please give a suggestion to help me work this exercise from Kunen's book.

Suppose $M$ satisfies the $CH$ and that we force over $M$ with $\mathbb{P}=Fn(I,2)$ where $(\omega_{2} \leq |I|))^{M}$, that is, with the finite partial functions from $I$ to $2$. If $f \in M[G] \cap \omega^{\omega}$, then is there necessarily a function $g \in M \cap \omega^{\omega}$ for which {$n:f(n) \leq g(n)$} is infinite? Please give a suggestion to help me work this exercise from Kunen's book.

supose $M$ satisface $CH$ and $\mathbb{P}=Fn(I,2)$ where $(\omega_{2} \leq |I|))^{M}$ if $f \in M[G] \cap \omega^{\omega}$ then existe $g \in M \cap \omega^{\omega}$ and {$n:f(n) \leq g(n)$} is infinite. I can work with this exercise Kunen book or give a suggestion, and that $(\omega_{2} \leq |I|))^{M}$.

Suppose $M$ satisfies the $CH$ and that we force over $M$ with $\mathbb{P}=Fn(I,2)$ where $(\omega_{2} \leq |I|))^{M}$, that is, with the finite-support functions from $I$ to $2$. If $f \in M[G] \cap \omega^{\omega}$, then is there necessarily a function $g \in M \cap \omega^{\omega}$ for which {$n:f(n) \leq g(n)$} is infinite? Please give a suggestion to help me work this exercise from Kunen's book.