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 $\omega$ such that

$A_0=A$;

$|A_{\alpha+1}\setminus A_\alpha|<\omega$; and

$|A_\alpha\setminus A_{\alpha+1}|=\omega$

for every $\alpha<\omega_1$?