Modifying a sequence of ordinals to be increasing and continuous. (To find names of small rank.) I want to prove the following:

Let $G$ be a generic subset of $\mathbb{P}=\text{Fn}(\mu,2)$ where $\mu$ is an ordinal, then for any ordinal $\alpha$ there exists $\beta\geq\alpha$ such that for any $x\in V[G]_\beta$ there is a name $\tau\in V_\beta$ such that $\tau_G=x.$

($\text{Fn}(\mu,2)$ is the set of finite partial functions $\mu\rightarrow 2$, i.e., Cohen forcing.)
As an attempt I define $\phi:V[G]\rightarrow\text{On}$ by
$$\phi(x) = \min\{\text{rk}(\tau)\mid \tau\in V^\mathbb{P}\text{ and }\tau_G=x\}
$$
and $\psi:\text{On}\rightarrow\text{On}$ by
$$
\psi(\xi)=\sup\{\phi(x)+1\mid x\in V[G]_\xi\}.
$$
Now, a fixed point of $\psi$ would give use the $\beta$ we are looking for. And a way to find a fixed point of $\psi$ would be to show it is continuous and strictly increasing. It is easily verified to be continuous, but I can't show it is increasing and it actually seems to me that it's probably not even true. Thus, the natural thing to do would be to modify $\psi$ slightly so that its fixed points still give us what we want but such that $\psi$ is strictly increasing. One such modification might be:
$$\psi(\xi)=\sup\{\phi(x)+\sup_{\alpha<\xi}\psi(\alpha)\mid x\in V[G]_\xi\}.$$ 
But I can't quite get it to work––it isn't neither clearly increasing nor clearly continuous.
This is an attempt to answer, at least partially, this question.
Thank you for any help!
 A: The argument you used (link) works. 
Let $G$ be $\mathbb{P}$-generic over $V$. Fix $\theta_{0} \in OR$ and consider 
\begin{gather*} \phi :V[G] \longrightarrow OR
\end{gather*}
where
\begin{gather*} \phi(x) = min\{ \beta \ | \ \exists \sigma \in (V_{\beta})^{V} ( \sigma_{G} = x ) \}
\end{gather*} and $\psi: OR \rightarrow OR$, where
\begin{gather*} \psi(\alpha) = sup\{ \phi(x) \ | \ x \in V_{\alpha}^{V[G]} \}
\end{gather*} Note that for all  $\alpha$ we have $\psi(\alpha) \geq \alpha  $.
Define $\langle \theta_{n} \ | \ n \leq \omega \rangle  $,  as follows:
\begin{gather*} \theta_{n+1} = \psi(\theta_{n}) + 1
\end{gather*}
and 
\begin{gather*} \theta_{\omega} = sup_{n\in\omega} \theta_{n}
\end{gather*}
It follows that $ n < m \leq \omega $ implies $\theta_{n} < \theta_{m}$.
(For $\beta \in OR$ we write $V_{\beta}[G]:= \{ \tau_{G} \ | \ \tau \in V_{\beta}  \ \& \ \tau \text{ is a name} \}$.) 
Fix $x \in V_{\theta_{\omega}}^{V[G]}$. Then $x \in V_{\theta_{n}}^{V[G]}$ for some $n < \omega$ and $V_{\theta_{n}}^{V[G]} \subseteq V_{\theta_{n+1}}[G] \subseteq V_{\theta_{\omega}}[G]$. 
