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Square Root

  1. Sep 20, 2011 #1
    The problem statement, all variables and given/known data

    If x > 0, then there exists a unique y > 0 such that y2 = x.

    The attempt at a solution

    Proof. Let A = {y ∈ Q : y2 < x}. A is bounded above by x, so lub(A) = η exists.
    Suppose η2 > x, where η = lub(A).
    Consider (η - 1/n)2 = η2 - 2η/n +1/n2 > η2 - 2η/n.
    Now η2 - 2η/n > x ⇔ η2 - x > 2η/n ⇔ (η2 - x)/2η > 1/n.
    We may choose such n by the Archmedean Property.
    Thus η - 1/n is an upper bound and η = lub(A), a contradiction.
    Similarly, if η2 < x, consider (η + 1/n)2 = η2 + 2η/n +1/n2 > η2 + 2η/n.
    Now η2 + 2η/n < x ⇔ 2η/n < x - η2 ⇔ 1/n < (x - η2)/2η.
    We may choose such n. So η is not an upper bound.
    Therefore, η2 = x by the Trichotomy rule. ∎
     
  2. jcsd
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