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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.
Template:Ordinal to word — does the same job as this template with "ord" set to 1; Template:Spellnum per MOS — wrapper of this template; applies guidance from WP:NUMERAL to determine when a number should be spelled out or not
The Turing degree of X ⊕ Y is the least upper bound of the degrees of X and Y. Thus is a join-semilattice. The least upper bound of degrees a and b is denoted a ∪ b. It is known that is not a lattice, as there are pairs of degrees with no greatest lower bound.
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).
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 ...
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See upper set. Upper bound. An upper bound of a subset X of a poset P is an element b of P, such that x ≤ b, for all x in X. The dual notion is called lower bound. Upper set. A subset X of a poset P is called an upper set if, for all elements x in X and p in P, x ≤ p implies that p is contained in X. The dual notion is called lower set.
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 ...