Definition. Let . We say is bounded above (below) if there exists such that for each , .
In this case, is called an upper bound ( is called a lower bound).
We say is bounded if is bounded above and below.
Remark. is possible.
Ex)
Show that has an upper bound and a lower bound (exercise).
ex) has a lower bound, but does not hove an upper bound.
ex) and
Then has a lower bound. .
However, does not have the maximum element.
To justify, it is enough to show if , then there exists such the . To do so, take any .
Take . Then .
Claim: .
.
Hence .
Definition. Let be bounded above We say is the least upper bound of if (1) is an upper bound of . (2) If , then is not an upper bound of . Denote called the Supremum of .
Remark.
(1) If sup E exists then SupE must be unique
Indeed, let be supremum of . Then either is not an upper bound. It is now esaily deduced that .
(2) Suppose is not bounded above. [There exists s.t for each For all , there exists S.t
(3) Let , not bounded above, .
(4) .
(5) Let be bounded above. there exists sup E by definition. Then for each , is not an upper bound. Thus by (2), there exists such that . Consequently, for each , there exists . Equivalently, .
(6) For each .
By (5) there exists such that .
In particular, this shows that we have “a sequence” ( satisfying In fact, (6) is equivalent to (5). This is called the Archimedean Property.