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Show that the set N* of finite sequences of nonnegative integers is countable.

Where do I start? I think I have to prove that there is a bijection between N* and N (set of natural numbers), but how do I get there?

Andrés E. Caicedo
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Adam Eisenstein
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1 Answers1

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Think about the function that sends a sequence $(a_1,a_2,a_3,\ldots,a_n)$ to $2^{a_1} 3^{a_2} 5^{a_3}\ldots p_n^{a_n}$, where $p_n$ denotes the $n$-th prime number.

Arthur
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  • Okay, but are those sets finite? They seem infinite, going up to an unspecified term. I just don't have enough of an idea how to do this yet. – Adam Eisenstein Oct 23 '13 at 23:52
  • What do you mean? The product of finitely many numbers is surely finite. – Arthur Oct 24 '13 at 00:02
  • @AdamEisenstein A set of the form ${a_1,\ldots,a_n}$, or ${p_1,\ldots p_n}$ for that matter, where $n$ is a natural number, is finite almost by definition. To be finite means to have a bijective correspondence with some natural number. – Trevor Wilson Oct 24 '13 at 00:03