Union of a finite set and a countably infinite set is countably infinite

Ok, here is the problem statement:

Prove that if $S$ is any finite set of real numbers, then the union of $S$ and the integers is countably infinite.

This seems pretty obvious to me, knowing that 2 countable sets are countable. But is there some step by step way to prove this? Like do I need to prove bijectivity or something? Thanks!

Solution 1:

Yes, give a bijection between $\Bbb N$ and $\Bbb Z\cup S$, that is a sequence in the latter which uses each element exactly once.

Say, $S\setminus\Bbb Z=\{s_1,..,s_k\}$. Then for example this sequence gives a bijection: $$(s_1,s_2,...,s_k,0,1,-1,2,-2,3,-3,4,-4,\dots)$$ That is, $1$ will be mapped to $s_1$, $k+1$ will be mapped to $0$, $k+2$ to $1$, and so on...

Solution 2:

It is simple to prove that the countable union of countable sets is countable: Lay out each set in a line (if finite, just repeat over and over), and then go over the elements $e_{i j}$ diagonally: First 0, 0; then 0, 1 and 1, 0; then 2, 0 and 1, 1 and 0, 2; ... If an element has shown up already, skip it. This gives a biyection between $\mathbb{N}$ and the union, unless the union is finite.