Why are natural numbers countable?

Definition - A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$.

Cantor's diagonal argument - Briefly, the Cantor's diagonal argument says:

Take $S = (0, 1) \subset \mathbb{R}$ and suppose that there exists an injective function $f$ from $S$ to $\mathbb{N}$. We prove that there exists an $s \in S$ (the diagonal number) such that $f(s)$ is not defined (i.e. there is no $n \in \mathbb{N}$ such that $f(s) = n$). Hence, $f$ is not an injective function and $S$ is uncountable.

Is $\mathbb{N}$ countable? - If you are concerned to prove this, just consider the function $$f(n) = n.$$ Obviolusly, this is a injective function since for every $n \in S = \mathbb{N}$, there is an $f(n)$ in $\mathbb{N}$, and viceversa. So, $\mathbb{N}$ is countable.

Is the set of even number countable? - Let $S$ be the even numbers set. Consider $$f(s) = \frac{s}{2}.$$ Again, this is a injective function, since for every even number $s \in S $, there is an $f(s) = \frac{s}{2}$ in $\mathbb{N}$, and viceversa. So, S is countable.

Conclusion - The Cantor's diagonal argument is just a method to prove that there is no injective function between two set. It's a tool to prove something, but it is not the unique method for proving countability.

Cantor's diagonal argument applied to any list of natural numbers written in decimal does indeed produce a decimal numeral not on the list.

A decimal numeral gives a natural number if and only if it repeats zeroes on the left; e.g. the number one is $\ldots 00001$.

So, the question you have to ask yourself is: why should the diagonal argument give a decimal numeral that repeats zeroes?