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The set of all bounded sequences forms the sequence space. [ citation needed ] The definition of boundedness can be generalized to functions f : X → Y {\displaystyle f:X\rightarrow Y} taking values in a more general space Y {\displaystyle Y} by requiring that the image f ( X ) {\displaystyle f(X)} is a bounded set in Y {\displaystyle Y} .
Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.
If also the sequence is bounded, then there are some real numbers and such that < and the sequence crosses the interval [,] infinitely often. That is, the sequence is eventually less than a {\displaystyle a} , and at a later time exceeds b {\displaystyle b} , and at an even later time is less than a {\displaystyle a} , and so forth ad infinitum.
Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain arbitrarily close to , meaning that given a real number greater than zero, all but a finite number of the elements of the sequence have a distance from less than .
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
The test is as follows. Let {g n} be a uniformly bounded sequence of real-valued continuous functions on a set E such that g n+1 (x) ≤ g n (x) for all x ∈ E and positive integers n, and let {f n} be a sequence of real-valued functions such that the series Σf n (x) converges uniformly on E. Then Σf n (x)g n (x) converges uniformly on E.
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a 1, a 2, a 3, ... satisfying . a n+p = a n. for all values of n. [1] [2] [3] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.