Analysis II | Part IB, 2003

Let (fn)n1\left(f_{n}\right)_{n \geqslant 1} be a sequence of continuous complex-valued functions defined on a set ECE \subseteq \mathbb{C}, and converging uniformly on EE to a function ff. Prove that ff is continuous on EE.

State the Weierstrass MM-test for uniform convergence of a series n=1un(z)\sum_{n=1}^{\infty} u_{n}(z) of complex-valued functions on a set EE.

Now let f(z)=n=1un(z)f(z)=\sum_{n=1}^{\infty} u_{n}(z), where

un(z)=n2sec(πz/2n).u_{n}(z)=n^{-2} \sec (\pi z / 2 n) .

Prove carefully that ff is continuous on C\Z\mathbb{C} \backslash \mathbb{Z}.

[You may assume the inequality coszcos(Rez)]|\cos z| \geqslant|\cos (\operatorname{Re} z)| \cdot]

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