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Definition: Bounded Subsets of Ordered Sets
Let $(V,\preceq )$ be a poset or a strictly ordered set and let $S\subseteq V$ be its non-empty subset.
1. \(S\) is called bounded above if it has an upper bound, i.e. there is an element $B\in V$ with $x\preceq B$ for all $x\in S.$
1. \(S\) is called bounded below if it has a lower bound, i.e. there is an element $B\in V$ with $x\succeq B$ for all $x\in S.$
If \(S\) is bounded above and bounded below, then we say that $S$ is bounded.
Examples and Notes
- Bounded subsets exist if we replace $V$ by the natural numbers $\mathbb N,$ the integers $\mathbb Z,$ the rational numbers $\mathbb Q,$ and the real numbers $\mathbb R,$ since they all are strictly ordered.
- If the underlying set is a poset, then only its chains can be bounded. If a subset of a poset is not a chain, then some of its elements might be not comparable with the bound in question, and then it does not make sense to speak about "the bound" of the subset.
Definitions: 2 3 4 5 6 7 8 9 10
Lemmas: 11 12
Proofs: 13 14 15 16 17 18 19 20
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- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983