Q:

Prove that the set of all periodic sequences (from some index) of natural numbers is countable

This exercise is from my course textbook and comes with a bunch of other exercises which practice the theorem that countable union of countable sets is countable. 
So I started by notating for every $k\in \mathbb{N}$ the set $X_k$ as the set of all periodic sequences from the index $k$ of natural numbers. now, $X=\displaystyle{\bigcup_{k\in \mathbb{N}}{X_k}}$ so it's only left to prove that for each $k\in \mathbb{N}$  the set $X_k$ is countable but I can't find an injection from $X_k$ to a countable set.

A:

Hint: Set $X_k^n$ to be the set of integer sequences that have period $k$, starting at index $n$. Now observe that $X = \bigcup \limits_{(k, n) \in \mathbb{N}^2} X_k^n$.

