Published Jun 17, 2021
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Axiom of Completeness. Every nonempty set of real numbers that is bounded above has a least upper bound
Definition 1.3.1A set \(A \subseteq B\) is bounded above if there exists a number \(b \in R\) such that \(a < b\) for all \(a \in A\). The number \(b\) is called an upper bound for \(A\).
Similarly, the set \(A\) is bounded below if there exists a lower bound \(l in R\) satisfying \(l < a\) for every \(a \in A\)
Definition 1.3.2 A real number \(s\) is the least upper bound for a set \(A \subseteq R\) if it meets the following two criteria
The least upper bound is also frequently called the supremum of the set \(A\). Although the notation \(s = lub A\) is sometimes used, we will always write \(s = sup A\) for the least upper bound.
The greatest lower bound or infimum for \(A\) is defined in a similar way and is denoted by \(inf A\)
Definition 1.3.4 A real number \(a_0\) is a maximum of set \(A\) if \(a_0\) is an element of A and \(a_0 \geqslant a\) for all \(a \in A\). Similarly, a number \(a_1\) is a minimum of \(A\) if \(a_1 \in A\) and \(a_1 \leqslant a\) for every \(a \in A\)
Let's turn our attention back to the Axiom of Completeness. Although we can see now that not every nonempty bounded set contains a maximum, the Axiom of Completeness asserts that every such set does have a least upper bound. An axiom in mathematics is an accepted assumption, to be used without proof. Preferably, an axiom should be an elementary statement about the system in question that is so fundamental that it seems to need to no justification.
Lemma 1.3.8 Assume \(s \in R\) is an upper bound for a set \(A \subseteq R\). Then, \(s = sup A\) if and only if, for every choice of \(\epsilon > 0\), there exists an element \(a \in A\) satisfying \(s - \epsilon < a\).