Prove that all finite sets are countable

Question

How can I prove that all finite sets are countable?

A set $S$ is countable if there exists an injective mapping $f:\mathbb{N}\rightarrow S$.

So, shall I prove the above by contradiction? I assume that there exists a finite set, which is not countable, i.e. there exists no injective mapping between the set and that of natural numbers. Then how to proceed to reach the contradiction?

Please help me solve this problem. Thanks in advance.

Answer 1

With your definition of a countable set no finite set is countable!

If $A$ is a finite set there exists a one-to-one function from $A$ into $\mathbb N$ and an onto function from $\mathbb N$ to $A$ but no bijective function exists.

Answer 2

By your definition of countable, finite sets aren't countable. If there is a bijection with $\Bbb N$, then $S$ is infinite.

The standard definition that I have seen of countable is that there exists an injective function $S\to \Bbb N$. In this case, finite sets are countable.

Answer 3

Under that definition of countable, finite sets are not countable. For instance, consider the set $\{1\}$. There is no bijection between this set and $\mathbb{N}$ as $\aleph_0=|\mathbb{N}|>|\{1\}|=1$