Cauchy's Convergence Criterion/General

Theorem

Let $\sequence {x_n}$ be a sequence in $\R$ or $\C$.


Then $\sequence {x_n}$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall r \in \N: r \ge N: \forall k > 0: \size {\ds \sum_{i \mathop = 1}^k x_{r + i} } < \epsilon$


Proof


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Source of Name

This entry was named for Augustin Louis Cauchy.


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy convergence condition: 2.
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy convergence condition: 2.
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