Real Numbers

Let $x \in \mathbb{R}$ be a real number. The absolute value or modulus of $x,$ written $\lvert x\rvert,$ is $x$ when $x$ is positive, $-x$ when $x$ is negative and $0$ when $x$ is zero.

$$\lvert x\rvert \, := \begin{cases} -x & x<0 \\ 0 & x=0 \\ x & x>0 \end{cases}$$

Let $x,y \in \mathbb{R}$ be real numbers. The distance between $x$ and $y,$ written $d(x,y),$ is the absolute value of their difference.

$$d(x,y) := \lvert x-y\rvert$$

Let $x,y \in \mathbb{R}$ be real numbers and $\varepsilon>0$ be a real number. The numbers $x$ and $y$ are $\boldsymbol{\varepsilon}$-close if and only if $d(x,y) \leq \varepsilon .$

Let $S \subseteq \mathbb{R}$ and $x \in \mathbb{R}.$ The number $x$ is an upper bound of $S$ if and only if $s \leq x$ for all $s \in S.$ The number $x$ is a lower bound of $S$ if and only if $x \leq s$ for all $s \in S.$

Let $S \subseteq \mathbb{R}.$ The set $S$ is bounded above if and only if it has an upper bound. The set $S$ is bounded below if and only if it has a lower bound. The set $S$ is bounded if and only if it is both bounded above and bounded below. Equivalently, $S$ is bounded if and only if $\exists M \geq 0 : \lvert s\rvert \leq M \, \forall s \in S.$

Let $S \subseteq \mathbb{R}$ and $\alpha \in \mathbb{R}.$ The number $\alpha$ is the supremum or least upper bound of $S,$ written $\sup S,$ if and only if $\alpha$ is an upper bound of $S$ and $\alpha \leq z$ for every upper bound $z$ of $S.$ The number $\alpha$ is the infimum or greatest lower bound of $S,$ written $\inf S,$ if and only if $\alpha$ is a lower bound of $S$ and $z \leq \alpha$ for every lower bound $z$ of $S.$ Every non-empty set that is bounded above has a supremum, and every non-empty set that is bounded below has an infimum.

Let $S \subseteq \mathbb{R}$ and $\alpha \in \mathbb{R}.$ The number $\alpha$ is the maximum of $S,$ written $\max S,$ if and only if $\alpha \in S$ and $s \leq \alpha$ for all $s \in S.$ The number $\alpha$ is the minimum of $S,$ written $\min S,$ if and only if $\alpha \in S$ and $\alpha \leq s$ for all $s \in S.$ The maximum and minimum have the following properties:

• if the maximum exists then it is the supremum, and if the minimum exists then it is the infimum;
• if $S=\varnothing$ or $S$ is not bounded above then the maximum does not exist;
• if $S=\varnothing$ or $S$ is not bounded below then the minimum does not exist;
• a non-empty bounded set need not have a maximum or minimum, for example the set $\{x\in\mathbb{R}:0<x<1\}$ has neither.

Let $X \subseteq \mathbb{R}.$ The set $X$ is an interval if and only if for all $x,y \in X$ with $x<y,$ every $z \in \mathbb{R}$ with $x<z<y$ is also in $X.$ The empty set is an interval since it has no such $x$ and $y.$

Let $X \subseteq \mathbb{R}$ be a non-empty bounded interval. The endpoints of $X$ are $a=\inf X$ and $b=\sup X.$ The interval $X$ is one of four sets, depending on which endpoints it contains:

• the open interval $(a,b):=\{x\in\mathbb{R}:a<x<b\};$
• the closed interval $[a,b]:=\{x\in\mathbb{R}:a\leq x\leq b\};$
• the half-open intervals $[a,b):=\{x\in\mathbb{R}:a\leq x<b\}$ and $(a,b]:=\{x\in\mathbb{R}:a<x\leq b\}.$

An unbounded interval is one of $(a,\infty),$ $[a,\infty),$ $(-\infty,b),$ $(-\infty,b]$ or $\mathbb{R}.$

Let $X \subseteq \mathbb{R}$ be a bounded interval. The length of $X,$ written $\lvert X\rvert,$ is $b-a$ if $X$ is non-empty with endpoints $a$ and $b,$ and $0$ if $X$ is empty.