(related to Proposition: Subset of a Countable Set is Countable)

- By hypothesis, $M$ is a countble and $A\subseteq M$ its subset.
- If $M$ is finite, then $A$ is finite, by the proposition about subsets of finite sets, and therefore countable.
- Assume $M$ is infinite.
- Since $A\subseteq M$, there is an injective function $f:A\to M.$
- By the comparison of cardinal numbers, it follows $|A|\le |M|$.
- Thus, $A$ is countable.∎

**Wille, D; Holz, M**: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994