Assume $CH$. $P$ is a countable support iteration of proper forcing notions of length $ \lt \omega_2$ and each forcing notion is forced to have size $ \le\omega_1$.
I can’t show $P$ forces $CH$. I tried to count the nice names for subsets of $\omega$,but I think the names are less than $|P|^{\omega\cdot |P|}=2^{\omega_1} $, it’s too large.
Please help me. I would appreciate any advice.