Subset of a finite set is finite We define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$.
How can we prove that a subset of a finite set is finite? It is of course sufficient to show that for a subset of $\{0,\ldots,n-1\}$. But how do I do that?

I have run into this old question and was surprised that noone seemed to have said the following.
The actual definition of finite set is the following: A set $A$ is finite if every injection $A\rightarrow A$ is a bijection.
(Note: this definition does not require the set $\mathbb{N}$)
Now let $B$ a finite set and $A\subset B$.
Suppose that $A$ is not finite. Then, by definition there exists a function
$f:A\rightarrow A$ that is injective but not surjective. Now define $F:B\rightarrow B$ as follows.
$$
F(x)=\begin{cases}
f(x) & \text{if $x\in A$}\\
x & \text{if $x\in B\setminus A$}
\end{cases}
$$
Clearly $F$ is injective but not surjective contradicting the finiteness of $B$.
