Let $X= \Pi_{n\in \omega} X_n\subset \Pi_{n\in \omega}\aleph_n$, where $X_n \subset \aleph_n$ is Lindelöf for each $n\in \omega$. Clearly, each $X_n$ has to be countable. My question is this:

Must $X$ be Lindelöf?

Thanks for any comment.