elementary set theory - proof: The set N of natural numbers is an infinite set

DEFINITION 1: A set $S$ is finite with cardinality $n \in\mathbb N$ if there is a bijection from the set $\left\{0, 1, ..., n-1 \right\}$ to $S$. A set is infinite if its not finite.

THEOREM 1: The set $\mathbb N$ of natural numbers is an infinite set.

Proof: Consider the injection $f :\mathbb N \rightarrow \mathbb N$ defined as $f(x) = 3x$. The range of $f$ is a subset of the domain of $f$.

I understand that $f(x) = 3x$ is not surjective and thus not bijective since for example the range does not contain number $2$. But what would happen if we were to define $f: \mathbb N\rightarrow \mathbb N$ as $f(x) = x$? It is a bijective function. Doesn't that make the set of natural numbers finite according to the definition? What am I missing can somebody please tell me?

Answer

No. The definition of finite is $f:\{0,1,...,n-1\}\to S$ is bijective.

We know $f:\mathbb N\to\mathbb N$ via $f(n) = n$ is bijective, but this maps $\mathbb N$ onto $\mathbb N$. It does not map $\{0,1,...,n-1\}$ onto $\mathbb N$.

Basically, this prove $\mathbb N$ is finite if $\mathbb N$ is finite.