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The plot of a convergent sequence {a n} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases. In the real numbers, a number is the limit of the sequence (), if the numbers in the sequence become closer and closer to , and not to any other number.
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 ...
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
The sequence () is said to be locally uniformly convergent with limit if is a metric space and for every , there exists an > such that () converges uniformly on (,). It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
Moreover, must be closed, since any limit point of , which has a sequence of points in converging to itself, must also lie in . Proof: (closed and bounded implies sequential compactness ) Since A {\displaystyle A} is bounded, any sequence { x n } ∈ A {\displaystyle \{x_{n}\}\in A} is also bounded.
One can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n approaches infinity of a sequence { a n } is simply the limit at infinity of a function a ( n ) —defined on the natural numbers { n } .
In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as n increases. In the real numbers every Cauchy sequence converges to some limit. A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large.