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Problem understanding a proof in Spivak Vol. 4

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data
    This is from Spivak, Vol. 4 Page 102-103

    Given |x-x_0| < 1, |x-x0| < Epsilon/(2(|y_0|+1))

    Also given |y-y_0| < Epsilon/(2(|x_0| + 1))

    Prove |xy-x_0y_0| < Epsilon


    2. Relevant equations
    See above


    3. The attempt at a solution

    The proof proceeds clearly enough. Using |x-x_0| < 1, he shows that |x| < |x_0| + 1.

    Then

    |xy-x_0y_0| = |x(y-y_0) + y_0(x-x_0)|

    < |x(y-y_0)| + |y_0(x-x_0)|

    < (1+|x_x0|)*Epsilon/(2(|x0|+1)) + |y_0|*Epsilon/(2(|y_0|+1))

    = Epsilon/2 + Epsilon/2 = Epsilon

    So.. Q.E.D., but I do not understand the second term...

    How is |y_0|*Epsilon/(2(|y_0| + 1)) = Epsilon/2 ??

    Any help would be most appreciated. This is for self-study, so I am without a teacher.

    Thanks,
    Shelly
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 11, 2011 #2
    That term, in fact, is not equal to [itex]\epsilon/2[/itex], it's merely less than it.

    Try verifying the inequality

    [tex] |y_0| \cdot \frac{ \epsilon}{2(|y_0|+1)} < \frac{\epsilon}{2}.[/tex]

    Spivak's book was my first introduction to rigorous calculus too. In these proofs I remember trying to equate everything (and having it never work). It's quite frustrating at first! Just remember analysis is all about inequalities.

    EDIT: Oh! Did you notice that that equal sign in the fourth line of your proof is supposed to be a less-than sign? Maybe that's where you got mixed up...?
     
    Last edited: Aug 11, 2011
  4. Aug 11, 2011 #3
    Stringy - thank you *so* much. That is a tremendous help.

    Much appreciated!

    Shelly
     
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