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Cauchy Sequences - Complex Analysis

  1. Jun 17, 2011 #1
    Hope someone could give me some help with a couple of problems.

    First:

    Proof of -

    A function f:G -->Complex Plane is continuous on G iff for every sequence C(going from 1 to infinity) of complex numbers in G that has a limit in G we have

    limit as n --> infinity f(C) = f(limit as n --> infinity C)

    Should I use the definition of the complex limit? Limit proofs always scare me.


    Next I need to provide a proof of "Every Cauchy sequence in the complex plane is convergent"

    I'm a little shaky on the definition of a Cauchy sequence if someone could provide one in more layman's terms that would be greatly appreciated.


    And lastly, were asked to "show that every Cauchy sequence of integers has a limit in the set of integers (Z)

    Again I feel with a better understanding of the definition of a Cauchy sequence I would have a shot at getting it.
     
  2. jcsd
  3. Jun 17, 2011 #2

    micromass

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    Hi MurraySt! :smile:

    I fear that you'll have to do some epsilon-delta things here. You'll need to prove

    [tex]\forall \varepsilon>0:\exists n_0:\forall n\geq n_0:~|f(x)-f(x_n)|<\varepsilon[/tex]

    Apply the definition of continuity here and the definition of [itex]x_n\rightarrow x[/itex].

    Do you know the corresponding theorem for the real numbers? I.e. do you know that R is complete? That should help you here.

    Intuitively, a Cauchy sequence is a sequence such that the terms lie closer and closer together. It are sequences that should converge in a space that's nice enough. And nice enough means complete here.

    Prove in general that a closed set of a complete space is complete.
     
  4. Jun 17, 2011 #3
    micromass gave a good intuitive definition, so I'll give a slightly more technical one. A sequence is Cauchy if, given any arbitrarily small positive number epsilon, we can find a point in the sequence after which all terms of the sequence are closer together than epsilon. Formally, for any [itex] \epsilon >0 [/itex] there is some N such that for any [itex] m,n \geq N [/itex], [itex]|s_m - s_n| < \epsilon[/itex].

    This seems like a little overkill for this problem (which is not to say it wouldn't work). The integers are a discrete set, so if we have, say |a - b| < 1/2 for integers a and b, what else can you say about a and b?
     
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