##  [Cauchy Criterion](/cauchy-criterion) 

  ##  [Cauchy Criterion](https://puremath.quantumdictionary.io/cauchy-criterion-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A characterization of convergence for sequences (or series via their sequence of partial sums) in metric or uniform spaces: a sequence is Cauchy if its terms become arbitrarily close to each other as the index increases; in a complete space, Cauchy sequences converge.