Exercise 14, § 5

General Topology, IV, 392

Let $(r_n)$ be an increasing sequence of finite real numbers $> 0$ , such that $\lim r_n = +\infty$ . For each finite real number $r > 0$ , let $N(r)$ denote the largest index $n$ such that $r_n \leq r$ . Show that

\limsup_{r \to \infty} \frac{N(r)}{r} = \limsup_{n \to \infty} \frac{n}{r_n}, \quad \liminf_{r \to \infty} \frac{N(r)}{r} = \liminf_{n \to \infty} \frac{n}{r_n}.

Solution

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