My question is the following:

I have an $\in$-chain of elementary submodels $\langle M_\xi\rangle_{\xi<\lambda^+}$ of $H_\theta$ for $\theta$ sufficiently large. I know that for any $M\prec H_\theta$, and for every $A\in M$ such that $A$ is countable in $M$, then $A\subseteq M.$ However, it would be great for my work if for every $A\in [\lambda^+]^\omega$, there is $\xi\in\lambda^+$ such that $A\in M_\xi.$ Do you know if it is true? and some easy arguments why yes, why not or why it depends?

Any good reference like "Everything you wanted to know about elementary submodels for set theory and you never dared to ask" would be also really appreciated.