Definition
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[22R] Let \(Aβ X\). The majorants of \(A\) (or upper bounds) are
\[ M_ A{\stackrel{.}{=}}\{ xβ X:β aβ A, aβ€ x\} \quad . \]
A set \(A\) is bounded above when there exists an \(xβ X\) such that \(β aβ A, aβ€ x\), i.e. exactly when \(M_ Aβ β \).
If \(M_ A\) has minimum \(s\), then \(s\) is th supremum, a.k.a. least upper bound, of \(A\), and we write \(s=\sup A\).
By reversing the order relation in the above definition, we obtain the definition of minorants/lower bounds, bounded below, infimum/greatest lower bound.