Question

(a) Show that a subset of a countable set is countable.

Answer 1

Let $B$ be a subset of a countable set $A$. By defenition of a countable subset, there is an injection $f:A\to \mathbb N$. Since the restriction $f|_B:B\to\mathbb N$ of an injection $f$ to the subset $B$ is an injection as well, we conclude that $B$ is a countable set.