 StatLect

Convergence criterion

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

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 ).

Example

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 obtain But converges to by increasing . Therefore, according to the convergence criterion above, the sequence converges to .