Let’s say we have a possibly infinite countable collection of countable sets , prove that
is countable too. We are going to do this using Axiom of Choice.
Since every set is countable, there is a set of injections from
to
, let’s call it
. Using the axiom of choice we can create a set of maps
from
to
. Now, we are going to create an injection from
to
. If
is an element of
then it’s also an element of some set
. Therefore we can construct
where
. And since we know that
is countable, we can map
to
. QED.