In http://www.proofwiki.org/wiki/Subset_of_Countable_Set [Broken] that subsets of a countable set are countable, by enumerating the elements of the subset with the labels n(adsbygoogle = window.adsbygoogle || []).push({}); _{i}, hasn't the author implicitly assumed the conclusion, namely that the subset is countable: that its elements can be labeled by the elements of [itex]\mathbb{N}[/itex]? However small one writes that i, tucked away in however many embedded levels of subscripts, it still stands for the numbers 1,2,3,..., doesn't it?

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# Subsets of a countable set

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