Let $A$ be an infinite subset of $\omega$ such that $\omega\setminus A$ is also infinite.

Under the Continuum Hypothesis is there a sequence $(A_\alpha)_{\alpha<\omega_1}$ of distinct subsets of $\omega$ such that $A_0=A$ and $|A_{\alpha+1}\setminus A_\alpha|<\omega$ for every $\alpha<\omega_1$?