Definition:
Lower and Greatest Lower Bound (Infimum)
Let A be an ordered set and X a subset of A. An
element b is called a lower bound for the set X if
every element in X is greater than or equal to b. If such a
lower bound exists, the set X is called bounded below.
Let A be an ordered set, and X a subset of A. An
element b in A is called a greatest lower bound
(or infimum) for X if b is a lower bound for X
and there is no other lower bound b' for X that is greater
than b. We write b = inf(X).
By its definition, if a greatest lower bound exists, it is unique.
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