Definition:Cauchy Sequence/Real Numbers
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Definition
Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$
Also see
- Definition:Complete Metric Space: a metric space in which the converse holds, that is a Cauchy sequence is convergent.
Thus in $\R$ a Cauchy sequence and a convergent sequence are equivalent concepts.
- Cauchy's Convergence Criterion on Real Numbers, where this is made explicit
Source of Name
This entry was named for Augustin Louis Cauchy.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Definition $1.2.8$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: $\S 5.16$: Cauchy sequences