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Convergent Sequences

  1. May 19, 2008 #1
    1. The problem statement, all variables and given/known data
    Let (X,d) be a metric space with two sequences [itex] (x_n), (y_n) [/itex] which converge to values of a,b respectively. Show that

    [tex] \lim_{n \to \infty} d(x_n,y_n) = d(a,b) [/tex]

    2. Relevant equations
    [tex] (x_n) \rightarrow a \Leftrightarrow \forall \epsilon >0 \quad \exists n_0 \in \mathbb{N} \text{ such that } \forall n>n_0 \quad d(x_n,a)< \epsilon [/tex]

    [tex] d(x,z) \leq d(x,y) + d(y,z) \quad \forall x,y,z \in X [/tex]

    3. The attempt at a solution

    This seems like it should be a fairly easy question, but I don't have much analysis in my background. I attempted to proceed as follows:

    Since [itex] d:X\times X \rightarrow \mathbb{R} [/itex], it is sufficient to show that [itex] \forall \epsilon >0 \quad \exists n\in \mathbb{N} \text{ such that } |d(x_n,y_n) - d(a,b)|< \epsilon [/itex]. So let [itex] \epsilon >0 [/itex] and [itex] n' = max\{ n_0, n_1 \} [/itex] where [itex] n_0, n_1 [/itex] are natural numbers which satisfy the individual convergence properties for [itex] (x_n),(y_n)[/itex]. Let [itex] n>n' [/itex] giving

    [tex] |d(x_n,y_n) - d(a,b) | &=& |d(x_n,y_n) + d(x_n,b) - d(x_n,b) - d(a,b)|

    \leq |d(y_n,b) - d(x_n,a)|

    < |\epsilon - \epsilon|

    \leq \epsilon[/tex]

    But I'm really not sure about the [itex] |\epsilon - \epsilon| \leq \epsilon [/itex] line.

    Any thoughts would be appreciated.
  2. jcsd
  3. May 19, 2008 #2


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    Well, if 0<=e1<=e and 0<=e2<=e then |e1-e2|<=e. That's what you really mean by |e-e|<=e, right?
  4. May 19, 2008 #3
    Yes, that was my thought process.
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