countable sets

A one-to-one correspondence of two sets means that there is a injective or surjective function between them. That is, two sets are one-to-one correspondent if and only if there is a into or onto mapping.

If one of two sets is the domain $\mathbb{N}$ and the other is the range $A=\left\{ a_n | n\in\mathbb{N}, a_n\in \mathbb{R} \right\}$, the function expresses a infinite real number sequence as follow.
\[ f : n\in\mathbb{N}\rightarrow a_n\in A \]
You may not suspect that the number of $a_n$ is countable with no limits.

[countable or uncountable]
A set is said to be countable, if it is finite or countably infinite. Otherwise, a set is uncountable.

In other words, if there is a one-to-one correspondence between $\mathbb{N}$ and a set $A$, then $A$ is countable. If a range set $A$ is finite, it bothers nobody. Even if the range set is infinite, it is possible for the set to be countable as a sequence noted above.

In this point of view, rational numbers are countably infinite. We will be able to accept it by making the sequence as follow.
\[  \frac{1}{1}, \frac{1}{2}, \frac{2}{2}, \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}, \cdots \]
This sequence is clearly countable and becomes rational numbers in addition to a plus or minus sign and zero. It may be most simple than any other proofs.