2.4. The Extended Real Line

Definition 2.40 (Extended real line)

The extended real number system or extended real line is obtained from the real number system $\mathbb{R}$ by adding two infinity elements $+\mathrm{\infty }$ and $-\mathrm{\infty }$, where the infinities are treated as actual numbers.

It is denoted as $\bar{\mathbb{R}}$ or $\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$.

The symbol $+\mathrm{\infty }$ is often written simply as $\mathrm{\infty }$.

In order to make $\bar{\mathbb{R}}$ a useful number system, we need to define the comparison and arithmetic rules of the new infinity symbols w.r.t. existing elements in $\mathbb{R}$ and between themselves.

2.4.1. Order

Definition 2.41 (Extended valued comparison rules)

We define the following rules of comparison between real numbers and infinities:

• $a<\mathrm{\infty }\mathrm{\forall }a\in \mathbb{R}$

• $a>-\mathrm{\infty }\mathrm{\forall }a\in \mathbb{R}$

• $-\mathrm{\infty }<\mathrm{\infty }$

In other words $-\mathrm{\infty }<a<\mathrm{\infty }\mathrm{\forall }a\in \mathbb{R}$.

Following notations are useful:

• $\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$

• $\mathbb{R}\cup \{\mathrm{\infty }\}=(-\mathrm{\infty },\mathrm{\infty }]$

• $\mathbb{R}\cup \{-\mathrm{\infty }\}=[-\mathrm{\infty },\mathrm{\infty })$

• $\mathbb{R}\cup \{-\mathrm{\infty },\mathrm{\infty }\}=[-\mathrm{\infty },\mathrm{\infty }]$

Definition 2.42 (Infimum and supremum in extended real line)

Let $A$ be a subset of $\mathbb{R}$.

• If $A$ is bounded from below, then $infA$ denotes its greatest lower bound.

• If $A$ is bounded from above, then $supA$ denotes its least upper bound.

• If $A$ is not bounded from below, we write: $infA=-\mathrm{\infty }$.

• If $A$ is not bounded from above, we write: $supA=\mathrm{\infty }$.

• For an empty set, we follow the convention as: $inf\varnothing =\mathrm{\infty }$ and $sup\varnothing =-\mathrm{\infty }$.

2.4.2. Arithmetic

Definition 2.43 (Extended valued arithmetic)

The arithmetic between real numbers and the infinite values is defined as below:

$$\begin{array}{r}\begin{array}{rl}& a+\mathrm{\infty }=\mathrm{\infty }+a=\mathrm{\infty }(-\mathrm{\infty }<a<\mathrm{\infty })\\ & a-\mathrm{\infty }=-\mathrm{\infty }+a=-\mathrm{\infty }(-\mathrm{\infty }<a<\mathrm{\infty })\\ & a\times \mathrm{\infty }=\mathrm{\infty }\times a=\mathrm{\infty }(0<a<\mathrm{\infty })\\ & a\times (-\mathrm{\infty })=(-\mathrm{\infty })\times a=-\mathrm{\infty }(0<a<\mathrm{\infty })\\ & a\times \mathrm{\infty }=\mathrm{\infty }\times a=-\mathrm{\infty }(-\mathrm{\infty }<a<0)\\ & a\times (-\mathrm{\infty })=(-\mathrm{\infty })\times a=\mathrm{\infty }(-\mathrm{\infty }<a<0)\\ & \frac{a}{\pm \mathrm{\infty }}=0(-\mathrm{\infty }<a<\mathrm{\infty })\end{array}\end{array}$$

The arithmetic between infinities is defined as follows:

$$\begin{array}{r}\begin{array}{rl}& \mathrm{\infty }+\mathrm{\infty }=\mathrm{\infty }\\ & (-\mathrm{\infty })+(-\mathrm{\infty })=-\mathrm{\infty }\\ & \mathrm{\infty }\times \mathrm{\infty }=\mathrm{\infty }\\ & (-\mathrm{\infty })\times (-\mathrm{\infty })=\mathrm{\infty }\\ & (-\mathrm{\infty })\times \mathrm{\infty }=-\mathrm{\infty }\\ & \mathrm{\infty }\times (-\mathrm{\infty })=-\mathrm{\infty }\end{array}\end{array}$$

Usually, multiplication of infinities with zero is left undefined. But for the purposes of mathematical analysis and optimization, it is useful to define as follows:

$$0\times \mathrm{\infty }=\mathrm{\infty }\times 0=0\times (-\mathrm{\infty })=(-\mathrm{\infty })\times 0=0.$$

2.4.3. Sequences, Series and Convergence

Definition 2.44 (Convergence to infinities)

A sequence $\{x_n\}$ of $\mathbb{R}$ converges to $\mathrm{\infty }$ if for every $M>0$, there exists $n_0$ (depending on M) such that $x_n>M$ for all $n>n_0$.

We denote this by:

$$lim{x}_n=\mathrm{\infty }.$$

A sequence $\{x_n\}$ of $\mathbb{R}$ converges to $-\mathrm{\infty }$ if for every $M<0$, there exists $n_0$ (depending on M) such that $x_n<M$ for all $n>n_0$.

We denote this by:

$$lim{x}_n=-\mathrm{\infty }.$$

We can reformulate Theorem 2.6 as:

Theorem 2.32 (Convergence of monotone sequences)

Every monotone sequence of real numbers converges to a number in $\bar{\mathbb{R}}$.

Proof. Let $\{x_n\}$ be an increasing sequence. If it is bounded then by Theorem 2.6, it converges to a real number.

Assume it to be unbounded (from above). Then, for every $M>0$, there exists $n_0$ (depending on M) such that $x_n>M$ for all $n>n_0$. Then, by Definition 2.44, it converges to $\mathrm{\infty }$.

Let $\{x_n\}$ be a decreasing sequence. If it is bounded then by Theorem 2.6, it converges to a real number.

Assume it to be unbounded (from below). Then, for every $M<0$, there exists $n_0$ (depending on M) such that $x_n<M$ for all $n>n_0$. Then, by Definition 2.44, it converges to $-\mathrm{\infty }$.

Thus, every monotone sequence either converges to a real number or it converges to one of the infinities.

Remark 2.13 (Infinite sums)

Consider a series $\sum x_n$. If the sequence of partial sums converges to $\mathrm{\infty }$, we say that $\sum x_n=\mathrm{\infty }$ i.e. the sum of the series is infinite. Similarly, if the sequence of partial sums converges to $-\mathrm{\infty }$, we say that $\sum x_n=-\mathrm{\infty }$.

Remark 2.14

Every series of non-negative real numbers converges in $\bar{\mathbb{R}}$.

Proof. The sequence of partial sums is an increasing sequence. By Theorem 2.32, it converges either to a real number or to $\mathrm{\infty }$.