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Definition D65
Cauchy sequence

Let $M = (X, d)$ be a D1107: Metric space.
A D62: Sequence $x : \mathbb{N} \to X$ is a Cauchy sequence with respect to $M$ if and only if $$\forall \, \varepsilon > 0 : \exists \, N \in \mathbb{N} \, (n, m \geq N \quad \implies \quad d(x_n, x_m) < \varepsilon)$$

Let $M = (X, d)$ be a D1107: Metric space.
A D62: Sequence $x : \mathbb{N} \to X$ is a Cauchy sequence in $M$ if and only if $$\forall \, \varepsilon > 0 : \exists \, N \in \mathbb{N} \, (n, m \geq N \quad \implies \quad x_m \in B_d(x_n, \varepsilon))$$
Children
 ▶ D724: Set of Cauchy sequences
Results
 ▶ R3228: Bounded sequence need not be Cauchy ▶ R244: Convergent sequence is Cauchy