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Cauchy sequence (Definition)

A sequence $ x_0, x_1, x_2, \dots$ in a metric space $ (X,d)$ is a Cauchy sequence if, for every real number $ \epsilon > 0$, there exists a natural number $ N$ such that $ d(x_n,x_m) < \epsilon$ whenever $ n,m > N$.

Likewise, a sequence $ v_0, v_1, v_2, \dots$ in a topological vector space $ V$ is a Cauchy sequence if and only if for every neighborhood $ U$ of $ \mathbf{0}$, there exists a natural number $ N$ such that $ v_n - v_m \in U$ for all $ n,m > N$. These two definitions are equivalent when the topology of $ V$ is induced by a metric.



"Cauchy sequence" is owned by djao. [ full author list (2) | owner history (1) ]
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See Also: metric space

Other names:  fundamental sequence

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if $d(x_i, x_{i+1})<1/2^i$ then $x_i$ is a Cauchy sequence (Result) by matte
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Cross-references: metric, induced, topology, equivalent, definitions, neighborhood, topological vector space, natural number, real number, metric space, sequence
There are 22 references to this entry.

This is version 5 of Cauchy sequence, born on 2001-10-27, modified 2005-11-30.
Object id is 600, canonical name is CauchySequence.
Accessed 18380 times total.

Classification:
AMS MSC: 26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line)
 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)
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