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

Aiming for a contradiction, suppose $S \setminus T$ is a finite set.

Then:

\(\ds S\)

\(\subseteq\)

\(\ds S \cup T\)

Set is Subset of Union

\(\ds \)

\(=\)

\(\ds \paren {S \setminus T} \cup T\)

Set Difference Union Second Set is Union

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.

$\blacksquare$