Analysis II | Part IB, 2005

Let (fn)n1\left(f_{n}\right)_{n \geqslant 1} be a sequence of continuous real-valued functions defined on a set ERE \subset \mathbf{R}. Suppose that the functions fnf_{n} converge uniformly to a function ff. Prove that ff is continuous on EE.

Show that the series n=11/n1+x\sum_{n=1}^{\infty} 1 / n^{1+x} defines a continuous function on the half-open interval (0,1](0,1].

[Hint: You may assume the convergence of standard series.]

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