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Convergent sequence meromorphic function proof periods discrete set Watch

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    The problem statement, all variables and given/known data

    Hi, As part of the proof that :

    the set of periods \Omega_f of periods of a meromorphic f: U \to \hat{C} , U an open set and \hat{C}=C \cup \infty , C the complex plane, form a discrete set of C when f is a non-constant

    a step taken in the proof (by contradiction) is : there exists an w_{0} \in \Omega_{f}  s.t for any open set U containing w_{0}, there is an w \in \Omega_{f} / {w_0} contained in U

    Now the next step is the bit I am stuck on

    By the standard trick in analysis, we can produce a sequence of periods \{w_n\} such that w_{n} \neq w_{0} and \lim_{n\to \infty} w_{n} = w_0

    The attempt at a solution

    It's been a few years since I've done analysis, and the 'trick' has no name so I am struggling to look it up and find it in google. A proof of this to understand it's meaning is really what I'm after , what's the idea behind the construction / significance in the usual context it would arise I am also confused with the notation, does n=0 ? So the sequence converges to it's first term, or is n starting from one

    many thanks
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    You've used U as an open set twice: once in the initial statement, and once in the "step taken in the proof bit". Are these the same sets, or different ones?

    If it would be equally correct to say:

    there exists an w_{0} \in \Omega_{f}  s.t for any open set W containing w_{0}, there is an w \in \Omega_{f} / {w_0} contained in W

    then it's basically trivial. (just take W_n to be the set \{z \in U:  |z-w_0| < \frac{1}{n} \}, then the corresponding w_n's form a sequence that works).

    If it's the same U, I'm not sure it's enough to deduce the result.

    In either case, I'm not sure what the "standard trick in analysis" is supposed to be here.
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    (Original post by DFranklin)

    then it's basically trivial. (just take W_n to be the set \{z \in U:  |z-w_0| < \frac{1}{n} \}, then the corresponding w_n's form a sequence that works).

    .
    thanks but I don't understand why the w_n generating by this are periods? So by our assumption we have f(w_n)=f(w_0) , for a single value of n. We have defined the open ball |w_n-w_0| < 1/n , if f was continous we could argue that we can find an \epsilon such that |f(w_n)-f(w_0)|<\epsilon and then taking the limit \epsilon \to 0 would give f(w_n)=f(w_0) the definition of a period, however holomorphic implies cts at every point but in this theorem the function f is mermorphic so this doesn't work?
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    (Original post by xfootiecrazeesarax)
    thanks but I don't understand why the w_n generating by this are periods?
    Each w_n is a member of \Omega_f, so it's a period, no?

    So by our assumption we have f(w_n)=f(w_0) , for a single value of n. We have defined the open ball |w_n-w_0| < 1/n , if f was continous we could argue that we can find an \epsilon such that |f(w_n)-f(w_0)|<\epsilon and then taking the limit \epsilon \to 0 would give f(w_n)=f(w_0) the definition of a period, however holomorphic implies cts at every point but in this theorem the function f is mermorphic so this doesn't work?
    None of this has anything to do with what you said you wanted to know. You said that:

    a step taken in the proof (by contradiction) is : there exists an s.t for any open set containing , there is an contained in

    Now the next step is the bit I am stuck on

    By the standard trick in analysis, we can produce a sequence of periods such that and
    I have explained to you why the 2nd statement follows from the first. Neither statement mentions f (besides the definition of \Omega_f) and f is not used or needed in any way in the proof.
 
 
 
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