Infinite if Injection from Natural Numbers

Theorem
Let $S$ be a set.

Let there exist an injection $\phi: \N \to S$ from the natural numbers to $S$.

Then $S$ is infinite.

Proof
that $S$ is finite.

Let $k \in \N$ be such that there exists a bijection $\psi: S \to \N_k$.

Note that $\N_k \subset \N$ since $k \in \N$, $k \notin \N_k$.

Consider the restriction $\phi \restriction \N_k$ of $\phi$ to $\N_k$.

Then $\phi \restriction \N_k: \N_k \to S$ is an injection by Restriction of Injection is Injection.

From Equivalence of Mappings between Finite Sets of Same Cardinality, it is also a bijection.

Consider $\map \phi k \in S$.

As $\phi \restriction \N_k$ is a surjection, there exists a $j \in \N_k$ such that:


 * $\map \phi j = \map {\phi \restriction \N_K} j = \map \phi k$

Since $j \ne k$, it follows that $\phi$ cannot be an injection.

So there can be no such $k$.

Hence $S$ is infinite.