A convergence criterion is a criterion used to verify the convergence of a sequence.

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The following is a commonly utilized convergence criterion.

Proposition Let be a sequence. Then, where is the distance between and .

In other words, a sequence is convergent to an element if and only if the terms of the sequence become closer and closer to when is increased (closeness is measured by a distance function ).

Consider a sequence of -dimensional vectors whose generic entry is

Because and converge to as becomes large, our intuition tells us that the sequence should converge to the vector defined as follows:How do we verify that this is indeed the case? First of all we need to define a distance function to measure the distance between and a generic term of the sequence . If we use Euclidean distance to measure the distance between and , we obtainBut converges to by increasing . Therefore, according to the convergence criterion above, the sequence converges to .

The lecture entitled Limit of a sequence provides more details about this convergence criterion.

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Please cite as:

Taboga, Marco (2021). "Convergence criterion", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/convergence-criterion.

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