Show that a countable union of countable sets is countable (take countable to mean either finite or countably infinite).
So I can see four cases:
Finite union of finite sets - this is easily proven to be countable
Countably infinite union of countably infinite sets - I've proved this by constructing a bijection to x
Finite union of countably infinite sets - subset of the above, hence countable as well
Countably infinite union of finite sets - the last one is the one I'm having problem with.
The hint I got was to use Schröder-Bernstein's theorem, but I'm having trouble constructing two injections for the required bijections.
Thanks in advance for any help!