One thought on “0.999…=1”

  1. Other proofs:

    1/3 = 0.333…
    2/3 = 2(1/3) = 2(0.333…) = 0.666…
    1 = 3/3 = 3(1/3) = 3(0.333…) = 0.999…

    Let x and y be distinct real numbers such that x < y, and that no real number z exists such that x < z < y. There must exist some real number a such that x + a = y. Since the real numbers are a field, this implies that a/2 is a real number, and since x + a/2 < x + a, there is a number z such that x < z < y. Contradiction, therefore x = y.

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