Question

Define a countable set and prove that countable union of countable set is countable.

Answer 1

Answer:

Theorem: Every countable union of countable sets is countable. We begin by proving a lemma; Lemma 1. A set X is countable if and only if there exists a surjection f : N → X. ... X is countable if and only if it is finite or denumerable. If X is denumerable, there is a bijection f : N → X.