Relative Difference between Infinite Set and Finite Set is Infinite

Theorem
Let $S$ be an infinite set.

Let $T$ be a finite set.

Then $S \setminus T$ is an infinite set.

Proof
$S \setminus T$ is a finite set.

Then:

By Union of Finite Sets is Finite, $S \cup T$ is a finite set.

By Subset of Finite Set is Finite, $S$ must also be a finite set.

But $S$ is an infinite set.

This is a contradiction.

Therefore $S \setminus T$ is an infinite set.