Definition:Cauchy Sequence/Real Numbers

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

• Convergent Sequence in Metric Space is Cauchy Sequence

• Definition:Complete Metric Space: a metric space in which the converse holds, that is a Cauchy sequence is convergent.

• Real Number Line is Complete Metric Space

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
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cauchy sequence