Editing Upper and lower bounds (section)

== Examples ==

For example, {{math|5}} is a lower bound for the set {{math|1=''S'' = {{mset|5, 8, 42, 34, 13934}}}} (as a subset of the [[integers]] or of the [[real numbers]], etc.), and so is {{math|4}}. On the other hand, {{math|6}} is not a lower bound for {{mvar|S}} since it is not smaller than every element in {{mvar|S}}. {{math|13934}} and other numbers ''x'' such that {{math|x ≥ 13934}} would be an upper bound for ''S''.

The set {{math|1=''S'' = {{mset|42}}}} has {{math|42}} as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that {{mvar|S}}.

Every subset of the [[natural number]]s has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the [[rational number]]s may or may not be bounded from below, and may or may not be bounded from above.

Every finite subset of a non-empty [[totally ordered set]] has both upper and lower bounds.