Exercise 10, § 7

General Topology, IV, 398

A series $(u_n)$ is convergent in $\mathbf{R}$ if and only if, for each increasing sequence $(p_n)$ of numbers $> 0$ such that $\lim_{n \to \infty} p_n = + \infty$ , we have

\lim_{n \to \infty} \left( \left( \sum_{k=0}^n p_k u_k \right) / p_n \right) = 0 \quad \text{(use Exercise 15 of § 5).}

Solution

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