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Have I proved this obvious fact correctly? (Real Analysis)

  1. May 5, 2012 #1
    1. The problem statement, all variables and given/known data
    It's not a HW problem. I was reading baby Rudin, in chapter 6 when it talks about Riemann–Stieltjes integral, it claims that given ε>0, we could choose η>0 such that (α(b)-α(a))η<ε. I wonder why it is true. I proposed this question to myself:

    Suppose that ε>0 is an arbitrarily given number. Is there a positive real number η such that for every real x we have: |x|η<ε?

    3. The attempt at a solution

    My solution is this: Suppose that such an η>0 doesn't exist. Therefore there exists a real number r such that for any η>0 we have: |r|η ≥ ε. Since η is now arbitrary, let's set η=1. we'll have 0<ε<|r|, but this restricts ε and that is in contradiction with the hypothesis that ε is arbitrarily given. (for example we could set ε=|r|+1 and that fails for sure).
     
  2. jcsd
  3. May 5, 2012 #2

    LCKurtz

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    I don't think that follows. Just do this: Given ##\eta## and ##\epsilon##, pick ##x>\frac \epsilon \eta##, which shows the statement is false.
     
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