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Math Help - complex numbers- hurwitz theorem

  1. #1
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    complex numbers- hurwitz theorem

    Suppose that D is a connected set open set , f_{n} \in H(D) and
    f_{n} -> f uniformly on compact subsets of D. If f is nonconstant and
     z \in D then there exists N and a sequence  z_{n} -> z such that  f_{n} ( z_{n} ) = f(z) for all n \geq N
    Hint: Assume that f(z) = 0. Apply the hurwitz theorem in the disk  D(z, r_{j}) for a suitable sequence  r_{j}->0

    I didn't even understand the hint.hurwitz theorem states that f_{n}
    and f have the same number of the zeroes in the disk. Can anyone help, thanx.
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  2. #2
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    Quote Originally Posted by hermanni View Post
    Suppose that D is a connected set open set , f_{n} \in H(D) and
    f_{n} -> f uniformly on compact subsets of D. If f is nonconstant and
     z \in D then there exists N and a sequence  z_{n} -> z such that  f_{n} ( z_{n} ) = f(z) for all n \geq N
    Hint: Assume that f(z) = 0. Apply the hurwitz theorem in the disk  D(z, r_{j}) for a suitable sequence  r_{j}->0

    I didn't even understand the hint.hurwitz theorem states that f_{n}
    and f have the same number of the zeroes in the disk. Can anyone help, thanx.
    I remember doing this exercise once and using a Cantor's diagonal argument somewhere, but I can't remember why I did all that. Take a look at this and let me know if something's off (It's pretty simple, which makes me wary):

    Pick z_0\in D with f(z_0)=0. There exists an r>0 such that f(z)\neq f(z_0)=0 for all z\in \overline{D}_{r}(z_0)\setminus \{ z_0 \} \subset D (the closed punctured disk or radius r and center z_0), and by Hurwiz theorem there exists N such that for all n\geq N the functions f_n have the same zeroes as f in the disk, so there exists z_n\in D_{r}(z_0) with f_n(z_n)=0. It now suffices to prove that z_n \rightarrow z_0, but this follows since \overline{D}_r(z_0) is compact (therefore sequentially compact) and the fact that the sequence z_n can only have one accumulation point by how we picked our disk and the fact that |f_n(y_n)-f(y)| \rightarrow 0 whenever y_n,y\in D and y_n\rightarrow y.

    Check it carefully, maybe I overlooked something.
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  3. #3
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    Everything seems OK to me , thank u very much )
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