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