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The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...
2 Examples. 3 Convergence of products. 4 See also. ... is bounded. Then is also ... is a strictly monotone and divergent sequence and the following limit exists:
The theorem states that each infinite bounded sequence in has a convergent subsequence. [1] An equivalent formulation is that a subset of R n {\displaystyle \mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded . [ 2 ]
A sequence that tends to a limit (i.e., exists) is said to be convergent; otherwise it is divergent. ( See the section on limits and convergence for details. ) A real-valued sequence ( a n ) {\displaystyle (a_{n})} is bounded if there exists M ∈ R {\displaystyle M\in \mathbb {R} } such that | a n | < M {\displaystyle |a_{n}|<M} for all n ∈ ...
There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence. Below are some of the more common and typical examples. This article is intended as an introduction aimed to help practitioners explore appropriate techniques.
a sequence = = is said to be Mackey convergent to the origin if there exists a divergent sequence = = of positive real numbers such that the sequence () = is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
If the integral of a function f is uniformly bounded over all intervals, and g is a non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral. Notes [ edit ]
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.