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