The subsets of finite sets are finite sets.

[evinda]

Hello! (Wave)

A set is called finite if it is equinumerous with a natural number $n \in \omega$.​

I want to show that the subsets of finite sets are finite sets.

That's what I have tried so far:

Let $A$ be a finite set.
Then $A \sim n$, for a natural number $n \in \omega$.
That means that there is a $1-1$ and surjective function $f: A \to n$.
Let $B \subset A$.
That means that $\forall x(x \in B \rightarrow x \in A)$.
What bijective function could we take in order to show that $B$ is also a finite set?

 

[Valkarie]

We can take the restriction of $f$ to $B$, $f|_B: B \to n$.Since $f$ is a bijective function, its restriction to $B$ is also a bijective function.Therefore, $B$ is equinumerous with $n$ and so $B$ is a finite set.