cauchy sequence in a sentence

Examples

1. Showing that a sequence is a Cauchy sequence is useful since we do not need to know the limit of the sequence in question.
2. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis.
3. Any Cauchy sequence of elements of " X " must be constant beyond some fixed point, and converges to the eventually repeating term.
4. 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.
5. :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.
6. 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.
7. 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.
8. 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.
9. In other words, a sequence is a Cauchy sequence if its elements " x " " n " eventually come and remain arbitrarily close to each other.
10. (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.