(sec:bra:extended-real-line)=
The Extended Real Line
=========================
```{prf:definition} Extended real line
:label: def-bra-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.
## Order
```{prf:definition} Extended valued comparison rules
:label: def-bra-erl-comparison
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:
- $\RR = (-\infty, \infty)$
- $\RR \cup \{ \infty\} = (-\infty, \infty]$
- $\RR \cup \{ -\infty\} = [-\infty, \infty)$
- $\RR \cup \{ -\infty, \infty\} = [-\infty, \infty]$
```{prf:definition} Infimum and supremum in extended real line
:label: def-bra-erl-infimum-supremum
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 $\sup \EmptySet = -\infty$.
```
## Arithmetic
```{prf:definition} Extended valued arithmetic
:label: def-bra-erl-arithmetic
The arithmetic between real numbers and the infinite values
is defined as below:
$$
\begin{aligned}
& a + \infty = \infty + a = \infty \quad (-\infty < a < \infty)\\
& a - \infty = -\infty + a = -\infty \quad (-\infty < a < \infty)\\
& a \times \infty = \infty \times a = \infty \quad (0 < a < \infty)\\
& a \times (-\infty) = (-\infty) \times a = -\infty \quad (0 < a < \infty)\\
& a \times \infty = \infty \times a = -\infty \quad (-\infty < a < 0)\\
& a \times (-\infty) = (-\infty) \times a = \infty \quad (-\infty < a < 0)\\
& \frac{a}{\pm \infty} = 0\quad (-\infty < a < \infty)
\end{aligned}
$$
The arithmetic between infinities is defined as follows:
$$
\begin{aligned}
&\infty + \infty = \infty\\
&(-\infty) + (-\infty) = -\infty\\
&\infty \times \infty = \infty\\
&(-\infty) \times (-\infty) = \infty\\
&(-\infty) \times \infty = -\infty\\
&\infty \times (-\infty) = -\infty
\end{aligned}
$$
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:
$$
0 \times \infty = \infty \times 0 = 0 \times (-\infty) = (-\infty) \times 0 = 0.
$$
```
## Sequences, Series and Convergence
```{prf:definition} Convergence to infinities
:label: def-bra-erl-convergence-infinity
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 {prf:ref}`res-bra-sequence-monotone-bounded-convergence` as:
```{prf:theorem} Convergence of monotone sequences
:label: res-bra-sequence-monotone-convergence
Every monotone sequence of real numbers converges to a number in $\ERL$.
```
```{prf:proof}
Let $\{x_n\}$ be an increasing sequence. If it is bounded then by
{prf:ref}`res-bra-sequence-monotone-bounded-convergence`, 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 {prf:ref}`def-bra-erl-convergence-infinity`, it converges to
$\infty$.
Let $\{x_n\}$ be a decreasing sequence. If it is bounded then by
{prf:ref}`res-bra-sequence-monotone-bounded-convergence`, 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 {prf:ref}`def-bra-erl-convergence-infinity`, it converges to
$-\infty$.
Thus, every monotone sequence either converges to a real number or it
converges to one of the infinities.
```
```{prf:remark} Infinite sums
:label: res-bra-erl-infinite-sums
Consider a {prf:ref}`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 $\sum x_n = -\infty$.
```
```{prf:remark}
:label: res-bra-erl-series-nng-convergence
Every series of non-negative real numbers converges in $\ERL$.
```
```{prf:proof}
The sequence of partial sums is an increasing sequence.
By {prf:ref}`res-bra-sequence-monotone-convergence`, it converges
either to a real number or to $\infty$.
```