If $B\subseteq\bigcup A$ is a countably infinite set, then $B_a=B\cap a$ is a Dedekind-finite set for every $a\in A$.
Show that every partition of a countably infinite set into Dedekind-finite sets is finite, and one of the parts has to be countably infinite.
Deduce that there is some $a\in A$ such that $B_a$ is countably infinite which is a contradiction to Dedekind-finiteness of $a$.
To prove the fact in the middle note that we may assume that the infinite set is $\mathbb N$ and that a partition is merely a surjection. Show, if so, that every surjection from $\mathbb N$ onto a Dedekind-finite set implies that the range is finite. (Hint: If $f\colon\mathbb N\to X$ is a surjection then there is an injection $g\colon X\to\mathbb N$.)