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A real-valued function is bounded if and only if it is bounded from above and below. [ 1 ] [ additional citation(s) needed ] An important special case is a bounded sequence , where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers .
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. [4]
A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.)
In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X). There are two common ways to define the limit of sequences of sets. In both cases:
An example is the chain complex defining the simplicial homology of a finite simplicial complex. A chain complex is bounded above if all modules above some fixed degree N are 0, and is bounded below if all modules below some fixed degree are 0. Clearly, a complex is bounded both above and below if and only if the complex is bounded.
The generalized Pareto distribution has a support which is either bounded below only, or bounded both above and below; The metalog distribution, which provides flexibility for unbounded, bounded, and semi-bounded support, is highly shape-flexible, has simple closed forms, and can be fit to data using linear least squares.
In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers ... converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums ...
2 Examples. 3 References. ... a uniformly bounded family of functions is a family of bounded functions that can all be ... The family of derivatives of the above ...