Superset of Infinite Set is Infinite

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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