1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Non-calc problem taken out from Spivak's Calculus, I reach a dead end

  1. Nov 18, 2013 #1
    1. The problem statement, all variables and given/known data

    The problem states that:

    [tex] y_0 \neq 0 [/tex]
    [tex] |y - y_0| < \frac{|y_0|}{2} [/tex]
    [tex] |y - y_0| < \frac{\epsilon|y_0|^2}{2} [/tex]

    And I am supposed to use these to prove that:

    [tex] y \neq 0 [/tex]
    [tex] |\frac{1}{y} - \frac{1}{y_0}| < \epsilon [/tex]

    2. Relevant equations

    [tex] |a| - |b| \leq |a - b| [/tex]
    (Reverse Triangle Inequality - my professor hinted me to use it)

    3. The attempt at a solution

    Ok proving the first part was easy, I simply used the reverse triangle inequality like this:

    1.I change [tex] |y - y_0| [/tex] to [tex] |y_0 - y| [/tex]
    2.I use the inequality: [tex] |y_0 - y| \geq |y_0| - |y| < \frac{|y_0|}{2} => -|y| < \frac{|y_0|}{2} - |y_0| => -|y| < -\frac{|y_0|}{2} => |y| > \frac{|y_0|}{2} [/tex] and since we know [tex] y_0 [/tex] is greater than 0, it's square and it's fractions are also greater than 0, so this means [tex] y [/tex] is greater than 0.

    Now for part 2 I played around with [tex] |\frac{1}{y} - \frac{1}{y_0}| [/tex] and even after transforming it by completing the fraction addition: [tex] |\frac{y_0 - y}{yy_0}| [/tex] I couldn't figure out how to proceed.Could someone help for this second part?
  2. jcsd
  3. Nov 18, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    For ##v > 0## the inequality ##|u| < v## is the same as ##-v < u < v##. Now just consider the two cases ##y_0 \geq 0## and ##y_0 < 0##.
  4. Nov 18, 2013 #3
    I thought [tex] y_o [/tex] could not be equal to 0.You mean [tex] y_0 > 0 [/tex] instead of [tex] y_0 \geq 0 [/tex] right?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted