Showing that a sequence is a Cauchy sequence is useful since we do not need to know the limit of the sequence in question.
42.
The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis.
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Any Cauchy sequence of elements of " X " must be constant beyond some fixed point, and converges to the eventually repeating term.
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Square integrability is equivalent to saying that a Cauchy sequence converges to a finite value under the weak topology : the space contains it's limit points.
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:Note that in these hypotheses, if the orbit of x 0 is bounded, then it is a Cauchy sequence and converges to a fixed point.
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Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on G, namely that for which H is a local base.
47.
The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance.
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A typical example is the set of real numbers, which may be defined through infinite decimal expansion, infinite binary expansion, Cauchy sequences, Dedekind cuts and many other ways.
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In other words, a sequence is a Cauchy sequence if its elements " x " " n " eventually come and remain arbitrarily close to each other.
50.
(This limit exists because the real numbers are complete . ) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0.
