Prove that the set of all finite subsets of $\mathbb{N}$ is countably infinite. I understand what countably infinite means but I don't know how to arrange the subsets according to their sum and I can't figure out how to form the proof Please help thanks
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1All subsets (as in the header) or all finite subsets (as in the body)? It might make a difference. – bof Nov 30 '17 at 21:20
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Each real number between $0$ and $1$ gives rise to a subset of $\mathbb N$ by truncating the decimal. For example, $\pi-3=.141592653\dots $ gives the subset ${1,14,141,1415,14159,141592,\cdots}$. Hence if you don't require finite the assertion is false. – lulu Nov 30 '17 at 21:22
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Map any finite sequence $(a_i){i=1}^n$ into $\prod{i=1}^n p_i^{a_i}$ where the $p_i$ are the primes. – marty cohen Nov 30 '17 at 22:01