Upper Bounds

If   SR,
then an upper bound for the set S
is a number   aR
such that
sa   for each   sS.

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We call a real number   y   a least upper bound of the set   S   if
(1)   y   is an upper bound for   S
and
(2)   if   a   is any upper bound for   S,   then   ya.

See ita-2-3-1-pg23.

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Lower Bounds

Analogous to the above there are the notions of lower bound and greatest lower bound:

is a lower bound for the subset if
for each ,

and a is greatest lower bound of S if
a is a lower bound of S and there exists no larger one.

S is called bounded from below if it has a lower bound.

See ita-2-3-8-pg25.

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