The absolute value or modulus \(\lvert x\rvert\) of a real number \(x \in \mathbb{R}\) 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 \(d(x,y)\) between \(x\) and \(y\) is the absolute value of their difference.

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

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

Let \(S \subseteq \mathbb{R}\) and \(x \in \mathbb{R}.\) \(x\) is an upper bound of \(S\) if \(s \leq x \, \forall s \in S\) and \(x\) is an lower bound of \(S\) if \(x \leq s \, \forall s \in S.\)

Let \(S \subseteq \mathbb{R}\) and \(\alpha \in \mathbb{R}.\) \(\alpha\) is a supremum or least upper bound of \(S\) if for all upper bounds \(z\) of \(S,\) \(\alpha \leq z.\) \(\alpha\) is an infimum or greatest lower bound of \(S\) if for all lower bounds \(z\) of \(S,\) \(z \leq \alpha.\)

Let \(S \subseteq \mathbb{R}\) and \(\alpha \in \mathbb{R}.\) \(\alpha\) is the maximum of \(S\) if \(\alpha \in S\) and \(s \leq \alpha \, \forall s \in S.\) \(\alpha\) is the minimum of \(S\) if \(\alpha \in S\) and \(\alpha \leq s \, \forall s \in S.\) If \(S=\varnothing\) or \(S\) is not bounded then the maximum or minimum does not exist.