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The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
Only a finite number of elements of the sequence are greater than this upper bound. The limit inferior of xn is the largest real number b that, for any positive real number \varepsilon, there exists a natural number N such that x_n>b-\varepsilon for all n > N. In other words, any number below the limit inferior is an eventual lower bound for ...
A subset of a poset is downward-directed if and only if its upper closure is a filter. Directed subsets are used in domain theory, which studies directed-complete partial orders. [5] These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of ...
A real number x is called an upper bound for S if x ≥ s for all s ∈ S. 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 ...
A real set with upper bounds and its supremum. A set S of real numbers is called bounded from above if there exists some real number k (not necessarily in S) such that k ≥ s for all s in S. The number k is called an upper bound of S. The terms bounded from below and lower bound are similarly defined. A set S is bounded if it
Bounded completeness can also be characterized differently. By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds. Consequently, bounded completeness is equivalent to the existence of all non-empty infima.
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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 ...