I'm reading about countable and uncountable sets, I found the following statement: "The set of the functions from [itex]\mathbb{Z}[/itex] to [itex][0,1][/itex] is uncountable" with the following proof: "To see that, suppose the set countable having the list [itex]\{f_1,f_2,\dots\}[/itex] and define [itex]f(x) = f_n(1/n)[/itex] if [itex]x=1/n[/itex] and [itex]f(x)=0[/itex] if [itex]x\neq 1/n[/itex] for any n".(adsbygoogle = window.adsbygoogle || []).push({});

Could someone explain this proof further?. It seems to me that he is trying to construct a function that is different from every [itex]f_i[/itex], but I don't see how the new function is necessarily different from every [itex]f_i[/itex], can't we have the possibilty that all the functions have the same values at [itex]1/n[/itex] for every [itex]n[/itex] but they are different for other values?.

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# Help understanding a proof

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