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.