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Homework Help: Equivalent metric functions

  1. Apr 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex]X = \textbf{b}[/tex] denote the set of all bounded real-valued
    sequences. Define the two metrics:
    [tex]
    \begin{align*}
    d_{\infty}(x,y) := \sup_{n \in \mathbb{N}} |x_n - y_n| \text{,
    and } d_e(x,y) := \sum^\infty_{n=1} \frac{1}{2^n} \frac{|x_n -
    y_n|}{1 + |x_n - y_n|}
    \end{align*}
    [/tex]
    for [tex]x = (x_1,x_2,\ldots), y = (y_1,y_2,\ldots) \in \textbf{b} = X[/tex].

    Prove that these metrics are not equivalent.

    2. Relevant equations

    For a space [tex] X \ne \varnothing [/tex], two distance functions [tex]d_1,d_2[/tex] are equivalent if for all sequences [tex]\{x_k \} \subset X[/tex] [tex]\lim_{k \to \infty} d_1(x_k,x) = 0[/tex] if and only if [tex]\lim_{k \to \infty} d_2(x_k,x) = 0[/tex].

    3. The attempt at a solution

    I guess the proof is to show that for sequence [tex]\boldsymbol{x}^{(k)}[/tex], [tex]\lim_{k \to \infty} d_{e}(x^{(k)}, x) \ne 0[/tex] when [tex]\lim_{k \to \infty} d_{\infty}(x^{(k)}, x) = 0[/tex]. But how can I prove this? What area of math do I need?
     
  2. jcsd
  3. Apr 9, 2010 #2

    LCKurtz

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    Think about {xn} where xn(k) = δnk, the Kronecker delta.
     
  4. Apr 9, 2010 #3
    Thank you for the suggestion. I can see this is a very smart counter example. Do you know with this kind of questions if we are supposed to prove by contradiction or a more generic way?
     
  5. Apr 9, 2010 #4

    LCKurtz

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    You yourself described above what you need to do. Try it with the sequence I suggested.
     
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