Question

Prove that the set of all real numbers R is uncountable.

Answer 1

Theorem. The real number sets is uncountable. x1 = f(1) y1 = f ( min{n ∈ N | x1 < f(n)} ) xn+1 = f ( min{n ∈ N | xn < f(n) < yn} ) yn+1 = f ( min{n ∈ N | xn+1 < f(n) < yn} ) . Then for every n ∈ N, we get xn < xn+1 < yn+1 < yn.

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