What is the sup of a sequence?
What is the sup of a sequence?
Given a sequence of real numbers , the supremum limit (also called the limit superior or upper limit), written and pronounced ‘lim-soup,’ is the limit of. as , where denotes the supremum. Note that, by definition, is nonincreasing and so either has a limit or tends to .
Is sup a limit point?
sup(A′) is not a limit point of A′.
What is the limit inferior of a function?
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function).
What does INF mean on calculator?
Contents
| Constant | Meaning |
|---|---|
| INF | Infinity or a real number larger than can be represented, e.g., 1/0 |
How to find the lim sup and lim inf?
If is the sequence of all rational numbers in the interval [0, 1], enumerated in any way, find the lim sup and lim inf of that sequence. The final statement relates lim sup and lim inf with our usual concept of limit. Conversely, if lim sup aj = lim inf aj are both finite then {aj} converges.
Which is the limit inferior of lim inf?
Let be a sequence of real numbers. Define Aj = inf {aj , aj + 1 , aj + 2 .} and let c = lim (Aj). Then c is called the limit inferior of the sequence . Let be a sequence of real numbers.
What happens if lim sup AJ and lim inf AJ converge?
Conversely, if lim sup aj = lim inf aj are both finite then {aj} converges. To see that even simple concepts like lim inf and lim sup can result in interesting math consider the following unproven conjecture:
Which is the outer limit of lim sup Xn?
lim sup Xn, which is also called the outer limit, consists of those elements which are limits of points in Xn taken from (countably) infinitely many n. That is, x ∈ lim sup Xn if and only if there exists a sequence of points xk and a subsequence { Xnk } of { Xn } such that xk ∈ Xnk and xk → x as k → ∞.
What is the sup of a sequence? Given a sequence of real numbers , the supremum limit (also called the limit superior or upper limit), written and pronounced ‘lim-soup,’ is the limit of. as , where denotes the supremum. Note that, by definition, is nonincreasing and so either has a limit or tends to .…