Greatest lower bounds in turn are given by the greatest common divisor.
2.
Bolzano was the first to recognize the greatest lower bound property of the real numbers.
3.
The Lebesgue outer measure emerges as the greatest lower bound ( infimum ) of the lengths from among all possible such sets.
4.
A set may have many lower bounds, or none at all, but can have at most one greatest lower bound.
5.
It is known that \ mathcal { D } is not a lattice, as there are pairs of degrees with no greatest lower bound.
6.
The symbols " and " denote the least upper bound and greatest lower bound operations, respectively, in the complete Boolean algebra " B ".
7.
That is, the least upper bound ( sup or supremum ) of the empty set is negative infinity, while the greatest lower bound ( inf or infimum ) is positive infinity.
8.
If it is over the whole of the real numbers then it may just have an infimum ( greatest lower bound ), but it may never attain it ( it could asymptotically approach it ).
9.
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.
10.
The notion of a complete lattice means that every subset of elements of has a unique least upper bound and a unique greatest lower bound; this generalizes the analogous upper bound and lower bound properties of the real numbers.
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