= Infimum and supremum
{title2=$\inf$, $\sup$}
{wiki}
For a <subset> $A$ of a <partially ordered set>, its <infimum> is its <greatest lower bound> and its <supremum> is its <least upper bound>. An <upper bound> only has to lie above every member; the <supremum> is the smallest such bound. A <lower bound> lies below every member; the <infimum> is the largest such bound. These definitions identify the exact edges of a <set> even when neither edge belongs to it.
For example, $(0,1)$ and $[0,1]$ both have <infimum> zero and <supremum> one. Only $[0,1]$ has a <minimum> and a <maximum>: those must be actual members of the <set>. The <set> $C=\{1/n:n\in\mathbb N_{>0}\}$ has <infimum> zero without a <minimum>, but its <supremum> one is also its <maximum>. Such examples distinguish a bound from an attained extremal value. <Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl/1.2 The set of real numbers>[Lebl, Basic Analysis I, §1.2: supremum, infimum and attainment]{ref}.
\Image[/infimum-supremum-attainment.png]
{title=<Infimum> and <supremum> need not be attained as a <minimum> or <maximum>}
{description=Filled endpoint markers belong to the displayed <set>; hollow markers do not. The reciprocal row shows selected values $1/n$, with further values accumulating toward the excluded lower edge zero. The finite row attains both bounds.}
For a nonempty <subset> of the <real numbers>, <completeness of the real numbers> guarantees a real <supremum> when it is bounded above, and a real <infimum> when it is bounded below. The bound and the ambient <partially ordered set> both matter: $\{q\in\mathbb Q:q\geq0,\ q^2<2\}$ has <supremum> $\sqrt2$ in the <real numbers>, but no <supremum> in the <rational numbers>. The detailed explanation belongs to <completeness of the real numbers>.
Why are unattained bounds useful? They specify a precise limiting target. In <mathematical optimization>, $f(x)=x$ on $(0,1)$ has <infimum> zero but no <global minimizer>. The inputs $x_n=1/n$ for $n\geq2$ give a <minimizing sequence>: their objective values approach zero without ever reaching it. Bounds therefore describe what approximation can achieve even when no feasible input is best. The <approximation property of an infimum> and <approximation property of a supremum> express this idea quantitatively.
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