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A question from Real and Complex Analysis (Rudin's).

  1. Mar 14, 2009 #1

    MathematicalPhysicist

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    I am trying to understand theorem 1.17 in page 15-16 international edition 1987.
    How do you show that [tex]\phi_n(t)[/tex] is a monotonic increasing sequence of functions?
     
  2. jcsd
  3. Mar 14, 2009 #2
    It might be easier if you note that

    [tex]k_n(t)=\text{floor}(2^nt)[/tex]

    and [tex]\text{floor}(2x)/2\ge \text{floor}(x)[/tex].

    Then, when you want to show that [tex]\varphi_n(t)\le\varphi_{n+1}(t)[/tex] consider the cases where [tex]0\le t<n[/tex], [tex]n\le t<n+1[/tex] and [tex]n+1\le t[/tex] separately.
     
  4. Mar 14, 2009 #3

    MathematicalPhysicist

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    Thanks, got it, basically I only need to check for t in [0,n) the other case is trivial.
     
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