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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 [eq1] be a sequence. Then, [eq2]where [eq3] is the distance between a_n and a.

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

Example

Consider a sequence [eq4] of $2$-dimensional vectors whose generic entry is[eq5]

Because $1/n$ and $2/n$ converge to 0 as n becomes large, our intuition tells us that the sequence [eq6] should converge to the vector a defined as follows:[eq7]How do we verify that this is indeed the case? First of all we need to define a distance function $d$ to measure the distance between a and a generic term of the sequence a_n. If we use Euclidean distance to measure the distance between a and a_n, we obtain[eq8]But $\sqrt{5}/n$ converges to 0 by increasing n. Therefore, according to the convergence criterion above, the sequence [eq9] converges to a.

More details

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

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