Lebl-Basic-Analysis-I · Problem 7.3.10

Metric Spaces

Mathematics · metric-spaces-definition

Problem

Take

\mathbb{R}^* = \{-\infty\} \cup \mathbb{R} \cup \{\infty\}

be the extended reals. Define

d(x, y) \colonequals \left|\frac{x}{1+|x|} - \frac{y}{1+|y|}\right|

x, y \in \mathbb{R}

, define

d(\infty, x) \colonequals \left|1 - \frac{x}{1+|x|}\right|

d(-\infty, x) \colonequals \left|1 + \frac{x}{1+|x|}\right|

for all

x \in \mathbb{R}

, and let

d(\infty, -\infty) \colonequals 2

. a) Show that

(\mathbb{R}^*, d)

is a metric space. b) Suppose

\{x_n\}_{n=1}^\infty

is a sequence of real numbers such that for every

M \in \mathbb{R}

, there exists an

N

such that

x_n \geq M

for all

n \geq N

. Show that

\lim_{n \to \infty} x_n = \infty

(\mathbb{R}^*, d)

. c) Show that a sequence of real numbers converges to a real number in

(\mathbb{R}^*, d)

if and only if it converges in

\mathbb{R}

with the standard metric.