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How can I rigorously prove that a finite set is countable?

A set is countable if there exists a bijective mapping from this set to that of natural numbers.

Please anyone help me to prove it. Thanks in advance.

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    $\begingroup$ Perhaps, it wants you to create a bijection from a fixed finite set and a subset of $\mathbb{N}.$ $\endgroup$
    – cmk
    May 20, 2019 at 20:46
  • $\begingroup$ If you say that a set is countable if it is finite there is nothing to prove. A finite set is countable by definition. If your definition of "countable" is different, please give details and we may be able to help. $\endgroup$ May 20, 2019 at 20:50
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    $\begingroup$ If "A set is countable if it is either finite or enumerable" is your definition of countable, then it is true by definition that a finite set is countable. $\endgroup$ May 20, 2019 at 20:56

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You will most likely want to appeal to the more rigorous definition of countability, namely that a set $S$ is countable if there is an injective function $f : S \to \mathbb{N}$. That is, you must show that every item in $S$ can be assigned to a unique natural number. Because $S$ is finite, this means that there are at most $k$ such elements in $S$. The rest of the "proof" is the trivial act of creating such an assignment for each element of $S$.

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