# Is the denumerable union of denumerable sets denumerable? [duplicate]

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?