The following is Exercise IV.7.10 in the 2013 edition of Kunen's "Set Theory":
Let $M$ be a countable, transitive model for ZFC. In $M$, let $\aleph_0 \le \kappa < \lambda$ be cardinals and consider the partial order $Fn(\kappa,\lambda)$ of function $f$ with finite domain $dom(f) \subset \kappa$ and $ran(f) \subset \lambda$ ordered via $f \le g :\Leftrightarrow f \supseteq g$.
Let $G$ be a $Fn(\kappa,\lambda)$-generic filter over $M$. Then in $M[G]$, $\lambda$ is countable and cardinals above $\lambda$ are preserved. Assuming that $M$ satisfies the generalized continuum hypothesis, show that $M[G]$ also satisfies GCH.
I was able to prove everything but $M[G] \models CH$ (sic!).
In fact, I can show that $M[G] \models 2^{\aleph_0} \le \lambda^{++} = \aleph_2$ using the standard arguments with nice names and $(Fn(\kappa,\lambda) \text{ has } \lambda^+\text{-cc})^M$. In order to prove $M[G] \models 2^{\aleph_0} = \aleph_1$ I tried to cook up a Surjection $\lambda^+ \rightarrow \mathcal P(\omega)$ in $M[G]$, but - so far - I couldn't figure out how to... Maybe someone can give me a hint?