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I was doing some homework exercises and I'm troubling with this problem. I let $\mathcal{F}:=\{S_1,S_2,\dots,S_n\}$ be any denumerable family, with $S_i$ is a denumerable set for all $i\in\mathbb{N}$, then I can write $S_i=\{s_{i1},s_{i2},\dots,s_{im}\}$ and exists $f_i:\mathbb{N}\mapsto S_i$ surjective for each $i$. Let $S=\bigcup_{n\in\mathbb{N}}$, I was trying to construct a function $f:\mathbb{N}\mapsto S$ and seeing that it's surjective too, but I don't know how to define it. I also know the other definition of denumerable, by having injective functions from $S_i$ to $\mathbb{N}$, should I use this definition instead?

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marked as duplicate by Scientifica, spaceisdarkgreen, Brahadeesh, Lord Shark the Unknown, Leucippus Nov 28 '18 at 6:27

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