Countable family of distinct infinite cardinalities with a largest member
= Countable family of distinct infinite cardinalities with a largest member
Pairwise distinct cardinalities do not prevent a countable family from having a largest member. For example, take $X_1=\aleph_\omega$ and $X_{n+2}=\aleph_n$ for $n<\omega$, viewing cardinals as initial ordinals. Every later set is a subset of $X_1$, so
$$
\bigcup_{n\geq1}X_n=X_1
$$
although all the cardinalities are different.