# Why is a set countable if there is a injective function?

For school, I have to prove that every finite subset of $\mathbb N$ is countable. Wikipedia tells me, that "By definition a set $S$ is countable if there exists an injective function $f$ from $S$ to the natural numbers.". I'm probably missing something obvious here, but why is this true?

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That means you can tag (or name, or count) each element in $S$ via an element in $\mathbb{N}$. That's actually the definition of counting. Which part exactly makes you uncomfortable in that sentence? – user13838 Oct 8 '11 at 8:41
@SrivatsanNarayanan Yes, but I didn't know ho to add the N... – fabian789 Oct 8 '11 at 8:47
@percusse mm, so you can say that $s_1$ is Element 1, $s_2$ Element 2? – fabian789 Oct 8 '11 at 8:49
Yes, exactly. If it is a finite element set, you can exhaust its elements by simply assigning elements from $\mathbb{N}$. That's what you are doing when counting anyway. :) Then, of course, there are infinite countable sets (odd numbers set etc.) and uncountable sets. – user13838 Oct 8 '11 at 8:56
If you write your comment as an answer I can accept it... – fabian789 Oct 8 '11 at 9:35

If it is a set with finite number of elements, you can exhaust its elements by simply assigning elements from $\mathbb{N}$. This is a special case of the definition : A set $S$ is countable if and only if there exists a one-to-one correspondence with a subset of natural numbers.