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Simple calculus proof

  1. Mar 13, 2006 #1
    I was supposed to have learnt this in my first year but i seem to have forgotten it because i havent kept in touch with it :mad:

    Definition:A sequence of complex numbers [itex] \left{z_{n}\right}_{1}^{\infty} [/itex] is said to have the limit Z0 or to converge to Zo and we write [tex] \lim_{n \rightarrow \infty} z_{n} = z_{0} [/tex] if for any epsilon>0 there exists an integer N such taht |Zn-Zo|< epsilon for all n>N
    Using the given definition prove that the sequence of complex number
    Zn = Xn + iYn converges to Zo = Xo + iYo iff Xn converges to Xo and Yn converges to Yo.
    [Hint: |Xn - Xo|<=|Zn - Zo|
    |Yn - Yo|<=|Zn-Zo|
    |Zn - Zo|<=|Xn - Xo|+|Yn - Yo|


    so we suppose the first part that
    Zn = Xn + iYn converges to Zo = Xo + iYo then Xn converges to Xo and Yn converges to Yo.

    well suppose it was triue then
    [tex] |Z_{n} - Z_{0}| = |X_{n} - X_{0} + iY_{n} - iY_{0}| \leq |X_{n} - X_{0}| + i|Y_{n} - Y_{0}| < \delta_{1} + \delta_{2} < \epsilon [/tex]

    not sure how to choose the pepsilon... do i make it the min of delta1 and delta 2 or the max?

    for the other way around i get that easily
    your help is greatly appreciated!
     
  2. jcsd
  3. Mar 14, 2006 #2

    benorin

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    You should not have an "i" in your inequality
     
  4. Mar 14, 2006 #3

    benorin

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    The iff will mandate a two-fold proof: the "if" part, and the "only if" part.

    Proof:

    The "if" part: If Zn = Xn + iYn converges to Zo = Xo + iYo, then Xn converges to Xo and Yn converges to Yo.

    Since Zn = Xn + iYn converges to Zo = Xo + iYo, we have

    [tex]\mbox{For every } \epsilon >0,\mbox{ there exists a }N\in\mathbb{N}\mbox{ such that }n>N\Rightarrow |z_n-z_0|<\epsilon[/tex]
     
  5. Mar 14, 2006 #4

    HallsofIvy

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    You do not "choose" epsilon. You have to show how to choose delta for any given epsilon.
     
  6. Mar 14, 2006 #5
    i am not really sure on how to use your advice...
    si does that mean the delta need to be replaces by epsilon1 and 2? Thereafter wechoose a delta that is the max of either?
     
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