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Spivak's Calculus (4th ed): Chapter 1 Problem *21 Inequality

  1. Nov 12, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that if
    ## |x-x_0|<\min (\frac {\epsilon}{2(|y_0|+1)},1)## and ##|y-y_0|<\frac{\epsilon}{2(|x_0|+1)} ##

    then
    ## |xy-x_0y_0|<\epsilon ##

    2. Relevant equations
    N/A

    3. The attempt at a solution
    From the first inequality I can see that:
    ## |x-x_0|<\frac {\epsilon}{2(|y_0|+1)} ## and ## |x-x_0|<1 ##
    From the first and second inequalities:
    ## |x-x_0|(2(|y_0|+1))<\epsilon ##
    ## |y-y_0|(2(|x_0|+1))<\epsilon ##
    so by adding up both inequalities I get:
    ## |x-x_0|+|y_0|\cdot|x-x_0|+|y-y_0|+|x_0|\cdot|y-y_0|<\epsilon ##
    and now I know that I need to write ##xy-x_0y_0## in a way that involves ##x-x_0## and ##y-y_0##

    Thanks in advance!
     
  2. jcsd
  3. Nov 12, 2015 #2

    Samy_A

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    Building on this, a hint:
    ##xy-x_0y_0=xy-xy_0+xy_0-x_0y_o=x(y-y_0)+y_0(x-x_0)##

    Now the ##|x||y-y_0|## term doesn't appear in your expression, but notice that ##x=x-x_0+x_o##, and remember that ##|x-x_0|<1##
     
    Last edited: Nov 12, 2015
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