438: Cauchy Sequence on Metric Space

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A definition of Cauchy sequence on metric space

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Topics

About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of metric space.

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Target Context

• The reader will have a definition of Cauchy sequence on metric space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

For any metric space, $M$, any sequence, $s:\mathbb{N}\to M$, where $\mathbb{N}$ is the positive natural numbers set, such that for any $ϵ\in \mathbb{R}$ such that $0<ϵ$, there is an $N\in \mathbb{N}$ such that for any $n_1,n_2\in \mathbb{N}$ such that $N<n_1,n_2$, $dist(s(n_1),s(n_2))<ϵ$

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2: Note

Sometimes, we may let $\mathbb{N}$ be the natural numbers set (with $0$).

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References

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