Let 2 be an open subset of C. Let f and (n)-1 be functions in H(22). Suppose that ( 1 is uniformly) bounded on compact subsets of 2, and that every subsequence of (n.1 that converges uniformly on compact subsets to some function, converges to f. Show that fn f uniformly on compact subsets of 12. Hint: Think first about the simpler situation where the functions are replaced by complex numbers. Imitate the same strategy.

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a The sequence zn defined by zn 1n converges to 0 To show that zn converges to 0 we need to show that for every positive number there exists a positiv... View the full answer