# 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 $$\RR$$ by adding two infinity elements $$+\infty$$ and $$-\infty$$, where the infinities are treated as actual numbers.

It is denoted as $$\ERL$$ or $$\RR \cup \{-\infty, +\infty\}$$.

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

In order to make $$\ERL$$ a useful number system, we need to define the comparison and arithmetic rules of the new infinity symbols w.r.t. existing elements in $$\RR$$ 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:

• $$a < \infty \Forall a \in \RR$$

• $$a > -\infty \Forall a \in \RR$$

• $$-\infty < \infty$$

In other words $$-\infty < a < \infty \Forall a \in \RR$$.

Following notations are useful:

• $$\RR = (-\infty, \infty)$$

• $$\RR \cup \{ \infty\} = (-\infty, \infty]$$

• $$\RR \cup \{ -\infty\} = [-\infty, \infty)$$

• $$\RR \cup \{ -\infty, \infty\} = [-\infty, \infty]$$

Definition 2.42 (Infimum and supremum in extended real line)

Let $$A$$ be a subset of $$\RR$$.

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

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

• If $$A$$ is not bounded from below, we write: $$\inf A = -\infty$$.

• If $$A$$ is not bounded from above, we write: $$\sup A = \infty$$.

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

## 2.4.2. Arithmetic#

Definition 2.43 (Extended valued arithmetic)

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

\begin{split} \begin{aligned} & a + \infty = \infty + a = \infty \quad (-\infty < a < \infty)\\ & a - \infty = -\infty + a = -\infty \quad (-\infty < a < \infty)\\ & a \times \infty = \infty \times a = \infty \quad (0 < a < \infty)\\ & a \times (-\infty) = (-\infty) \times a = -\infty \quad (0 < a < \infty)\\ & a \times \infty = \infty \times a = -\infty \quad (-\infty < a < 0)\\ & a \times (-\infty) = (-\infty) \times a = \infty \quad (-\infty < a < 0)\\ & \frac{a}{\pm \infty} = 0\quad (-\infty < a < \infty) \end{aligned} \end{split}

The arithmetic between infinities is defined as follows:

\begin{split} \begin{aligned} &\infty + \infty = \infty\\ &(-\infty) + (-\infty) = -\infty\\ &\infty \times \infty = \infty\\ &(-\infty) \times (-\infty) = \infty\\ &(-\infty) \times \infty = -\infty\\ &\infty \times (-\infty) = -\infty \end{aligned} \end{split}

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:

$0 \times \infty = \infty \times 0 = 0 \times (-\infty) = (-\infty) \times 0 = 0.$

## 2.4.3. Sequences, Series and Convergence#

Definition 2.44 (Convergence to infinities)

A sequence $$\{ x_n\}$$ of $$\RR$$ converges to $$\infty$$ 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:

$\lim x_n = \infty.$

A sequence $$\{ x_n\}$$ of $$\RR$$ converges to $$-\infty$$ 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:

$\lim x_n = -\infty.$

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 $$\ERL$$.

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 $$\infty$$.

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 $$-\infty$$.

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

Remark 2.14

Every series of non-negative real numbers converges in $$\ERL$$.

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