1.II.11B

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

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

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

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