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$\mathrm{\left[}\mathrm{\right]}$ Let $\mathit{S}$ be any non‐empty subset of $\mathbb{R}$. The set of

$\mathrm{•}$ upper bounds of $\mathit{S}$ is $\mathit{U}\mathrm{\left(}\mathit{S}\mathrm{\right)}\mathrm{=}\mathrm{\left\{}\mathit{u}\mathrm{\in }\mathbb{R}\mathrm{ }\mathrm{:}\mathrm{ }\mathit{u}\mathrm{\ge }\mathit{s}\mathrm{\forall }\mathit{s}\mathrm{\in }\mathit{S}\mathrm{\right\}}\mathrm{.}$

$\mathrm{•}$ lower bounds of $\mathit{S}$ is $\mathit{L}\mathrm{\left(}\mathit{S}\mathrm{\right)}\mathrm{=}\mathrm{\left\{}\mathit{l}\mathrm{\in }\mathbb{R}\mathrm{ }\mathrm{:}\mathrm{ }\mathit{l}\mathrm{\le }\mathit{s}\mathrm{\forall }\mathit{s}\mathrm{\in }\mathit{S}\mathrm{\right\}}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Let $\mathit{S}$ be any non‐empty subset of $\mathbb{R}$. Then

$\mathrm{•}$ the supremum of $\mathit{S}$ in $\mathbb{R}$ is the unique smallest element in $\mathit{U}\mathrm{\left(}\mathit{S}\mathrm{\right)}$ .

$\mathrm{•}$ the infimum of $\mathit{S}$ in $\mathbb{R}$ is the unique largest element in $\mathit{L}\mathrm{\left(}\mathit{S}\mathrm{\right)}$ .

$\mathrm{•}$ the supremum and infimum of $\mathit{S}$ always exist.

$\mathrm{•}$ the supremum (infimum) is also referred to as the least upper bound (greatest lower bound).

$\mathrm{\left[}\mathrm{\right]}$ Let $\mathit{S}$ be any non‐empty subset of $\mathbb{R}$. Then

$\mathrm{•}$ a maximum of $\mathit{S}$ in $\mathbb{R}$ is an element ${\mathit{s}}^{\mathrm{*}}\mathrm{\in }\mathit{S}$ with ${\mathit{s}}^{\mathrm{*}}\mathrm{\ge }\mathit{s}\mathrm{\forall }\mathit{s}\mathrm{\in }\mathit{S}\mathrm{.}$

$\mathrm{•}$ a minimum of $\mathit{S}$ in $\mathbb{R}$ is an element ${\mathit{s}}_{\mathrm{*}}\mathrm{\in }\mathit{S}$ with ${\mathit{s}}_{\mathrm{*}}\mathrm{\le }\mathit{s}\mathrm{\forall }\mathit{s}\mathrm{\in }\mathit{S}\mathrm{.}$

$\mathrm{•}$ a maximim and/or minimum of $\mathit{S}$ may not exist.

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