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 2.5 as:

Theorem 2.31 (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 2.5, 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 2.5, 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 2.31, it converges either to a real number or to \(\infty\).