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A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers .
If (,) is a partially ordered set, such that each pair of elements in has a meet, then indeed = if and only if , since in the latter case indeed is a lower bound of , and since is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original ...
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
On the other hand, / is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal between / and , and if / < / then is not infinitesimal. But 1 / ( 4 n ) < c / 2 {\displaystyle 1/(4n)<c/2} , so c / 2 {\displaystyle c/2} is not infinitesimal, and this is a contradiction.
For example, the least upper bound of a bounded increasing computable sequence of computable real numbers need not be a computable real number. [9] A sequence with this property is known as a Specker sequence , as the first construction is due to Ernst Specker in 1949. [ 10 ]
The construction follows a recursion by starting with any number , that is not an upper bound (e.g. =, where and an arbitrary upper bound of ). Given I n = [ a n , b n ] {\displaystyle I_{n}=[a_{n},b_{n}]} for some n ∈ N {\displaystyle n\in \mathbb {N} } one can compute the midpoint m n := a n + b n 2 {\displaystyle m_{n}:={\frac {a_{n}+b_{n ...