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Suppose each of your countable sets actually is $\mathbb{N}$. Then a countable product of them would have elements of the form $$(n_0,n_1,\dots)$$ i.e. an infinite sequence of natural numbers. Any real number in $[0,1]$ has a decimal expansion of the form $$0.d_0d_1\dots$$ i.e. an infinite sequence of digits in $\{0,\dots,9\}$. Since $\{0,\dots,9\}\...



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