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Proof

(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.
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References

Bibliography

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