Please see below this example from my notes. I basically understand it except for where it defines the function $f$. If I am not mistaken shouldn’t it be $f(\xi):= \xi + n, (\xi \in \omega \setminus n)$ because as $n\leqslant \omega$ then $\xi \in \omega$ could also make $\xi \in n$?
Also on a general note why does it define $f$ for $\xi \in n$ and not $\xi \leqslant n$ and similar for the others?
