Let $X= \prod_{n\in \omega} X_n\subset \prod_{n\in \omega}\aleph_n$, where $X_n \subset \aleph_n$ (where $\aleph_n$ is the space with order topology) is Lindelöf for each $n\in \omega$. My question is this:

Must $X$ be Lindelöf?

Thanks for any comment.