View source for Hurwitz theorem

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Let $(f_n(z))$ be a sequence of [[holomorphic function]]s in a domain $D \subset \mathbb{C}$ that converges uniformly on compact sets in $D$  to a function $f(z) \not\equiv 0$. Then, for any closed [[Rectifiable curve|rectifiable]] [[Jordan curve]] $\Gamma$ lying in $D$ together with the domain bounded by $\Gamma$ and not passing through zeros of $f(z)$, it is possible to find a number $N = N(\Gamma)$ such that for $n > N$ each of the functions $f_n(z)$ has inside $\Gamma$ the same number of zeros as $f(z)$ inside $\Gamma$. Obtained by A. Hurwitz .

====References====
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<TR><TD valign="top">[1a]</TD> <TD valign="top">  A. Hurwitz,   "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt"  ''Math. Ann.'' , '''46'''  (1895)  pp. 273–284</TD></TR>
<TR><TD valign="top">[1b]</TD> <TD valign="top">  A. Hurwitz,   "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" , ''Math. Werke'' , '''2''' , Birkhäuser  (1933)  pp. 533–545</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR>
</table>



====Comments====
For another theorem using  "nearness of functions"  to derive  "equality of number of zeros"  see [[Rouché theorem|Rouché's theorem]].