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Homework Help: (Tricky) Absolute Value Inequalities

  1. Feb 13, 2012 #1
    Hello everyone,

    I'm posting here since I'm only having trouble with an intermediate step in proving that

    [tex] \sqrt{x} \text{ is uniformly continuous on } [0, \infty] [/tex].


    By definition, [tex] |x - x_0| < ε^2 \Longleftrightarrow -ε^2 < x - x_0 < ε^2 \Longleftrightarrow -ε^2 + x_0 < x < ε^2 + x_0 [/tex]

    1. How does this imply the inequality in red?

    [tex] \text{ Since } ε > 0 \text{ then } x_0 - ε^2 < x_0 [/tex]

    However, I do not know more about x0 vs x.

    2. Also, how does the above imply the case involving the orange; what "else" is there?

    Thank you very much!
    Last edited: Feb 13, 2012
  2. jcsd
  3. Feb 13, 2012 #2


    Staff: Mentor

    The inequality |x - x0| < ε2 doesn't specify whether x is to the right of x0 or to the left of it. That's the reason for the two inequalities.
  4. Feb 13, 2012 #3
    Thank you for your response, Mark44.

    Could you please explain the red box?
  5. Feb 13, 2012 #4


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    Staff Emeritus
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    It looks like that's exactly what he did !
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