Infinite Set has Countably Infinite Subset/Proof 1

Theorem
Every infinite set has a countably infinite subset.

Proof
Let $S$ be an infinite set.

We use Between Two Sets Exists Injection or Surjection.

Suppose that there exists an injection $\psi: \N \to S$.

Let $T$ be the image of $\psi$.

From Injection to Image is Bijection, it follows that $\psi^{-1}: T \to \N$ is a bijection.

Hence, $T$ is a countably infinite subset of $S$.

Now, suppose that that there exists a surjection $\phi: \N \to S$.

From Surjection from Natural Numbers iff Countable, it follows that $S$ is countably infinite.

So, from Set is Subset of Itself, we have that $S$ is a countably infinite subset of $S$.