Superset of Infinite Set is Infinite

Theorem

Let $S$ be an infinite set.

Let $T \supseteq S$ be a superset of $S$.

Then $T$ is also infinite.

Proof

Suppose $T$ were finite.

Then by Set Finite iff Injection to Initial Segment of Natural Numbers, there is an injection $f: T \to \N_{<n}$ for some $n \in \N$.

But then by Restriction of Injection is Injection, also the restriction of $f$ to $S$:

$f {\restriction_S}: S \to \N_{<n}$

is an injection.

Again by Set Finite iff Injection to Initial Segment of Natural Numbers, this contradicts the assumption that $S$ is infinite.

Hence $T$ is infinite.

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 6$: Finite Sets: Exercise $2$