Question

Prove that a countable union of countable sets countable; i.e., if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai countable, then union i∈I Ai is countable. Hints: (i) Show that it suffices to prove this for the case in which I = N and, for every i ∈ N, the set Ai is nonempty. (ii) In the case above, a result proven in class shows that for each i ∈ N there is a surjective map fi : N → Ai. Use these maps to produce a surjective mapN×N → union i∈N Ai, and then use earlier results to conclude that union i∈N Ai is countable.

Answer 1