Basic Analysis I — Real Analysis · Problem 7.3.10

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · sequences-and-convergence

Problem

Take R∗={−∞}∪R∪{∞}\mathbb{R}^* = \{-\infty\} \cup \mathbb{R} \cup \{\infty\} be the extended reals. Define d(x,y):−∣x1+∣x∣−y1+∣y∣∣d(x,y) \coloneq \left|\frac{x}{1+|x|} - \frac{y}{1+|y|}\right| if x,y∈Rx, y \in \mathbb{R}, define d(∞,x):−∣1−x1+∣x∣∣d(\infty, x) \coloneq \left|1 - \frac{x}{1+|x|}\right|, d(−∞,x):−∣1+x1+∣x∣∣d(-\infty, x) \coloneq \left|1 + \frac{x}{1+|x|}\right| for all x∈Rx \in \mathbb{R}, and let d(∞,−∞):−2d(\infty, -\infty) \coloneq 2. a) Show that (R∗,d)(\mathbb{R}^*, d) is a metric space. b) Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is a sequence of real numbers such that for every M∈RM \in \mathbb{R}, there exists an NN such that xn≥Mx_n \geq M for all n≥Nn \geq N. Show that lim⁡n→∞xn=∞\lim_{n\to\infty} x_n = \infty in (R∗,d)(\mathbb{R}^*, d). c) Show that a sequence of real numbers converges to a real number in (R∗,d)(\mathbb{R}^*, d) if and only if it converges in R\mathbb{R} with the standard metric.

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