Bounds Definitions
Some definitions worth keeping in mind while studying for the next of proofs.
Axiom of Completeness
Upper and Lower Bounds
Note here that \(b\) doesn’t need to be \(A\). A set like \(\{1,2\}\) can have an upper bound like 3 for example. Every element in this set is less than 3. Also note here that \(A\) and \(b\) are both in \(\mathbf{R}\).
Questions: What if we don’t find such a \(b\)? can we conclude that the set is infinite?
Least Upper and Greater Lower Bounds
(i) \(s\) is an upper bound for A;
(ii) if \(b\) is any upper bound for \(A\), then \(s \leq b\).
The least upper bound is also called the supremum of the set \(A\) or \(\sup A\) while the greatest lower bound is called the infimum or \(\inf A\). Both bounds are unique. There can only be least upper bound and one greatest lower bound.
Questions: since \(s\) must be an upper bound, does it mean that the set \(A\) is a bounded set?
Minimum and Maximum
It’s definitely clear here that the maximum and minimum must be elements of the set itself unlike the upper and lower bounds.