In this work, we take a closer look at the problem of convergence of sequences of holomorphic functions. It is in our interest to specify the settings in which the limit function is holomorphic and provide examples in which the limit function fails to have the desired property. We construct such examples primarily with the use of Runge's theorem. We also study theorems which classify the problem of convergence with a special emphasis on Osgood's theorem and its consequences.
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