Definition:Cauchy Sequence/Complex Numbers
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Definition
Let $\sequence {z_n}$ be a sequence in $\C$.
Then $\sequence {z_n}$ is a Cauchy sequence if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {z_n - z_m} < \epsilon$
where $\size {z_n - z_m}$ denotes the complex modulus of $z_n - z_m$.
Considering the complex plane as a metric space, it is clear that this is a special case of the definition for a metric space.
Also see
- Definition:Complete Metric Space: a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.
Thus in $\C$ a Cauchy sequence and a convergent sequence are equivalent concepts.
Source of Name
This entry was named for Augustin Louis Cauchy.
Sources
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cauchy sequence