View source for Equivalence of Cauchy sequences

==Definition==
{{:Equivalence of Cauchy sequences/Definition}}<ref group="Note">In [[Krzysztof Maurin's notation]] this would be written as:
* {{MM|1=\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{n>N}d(a_n,b_n)<\epsilon}}</ref>
We then write {{M|(a_n)\sim(b_n)}} (we often omit the {{M|1=n=1}} and such as [[mathematicians are lazy]]) and denote the [[equivalence class|equivalence classes]] as {{M|[(a_n)]}} or even just {{M|[a_n]}} (provided this is unambiguous in the context)
==Proof of claim==
{{Begin Inline Theorem}}
'''[[:Equivalence of Cauchy sequences/Proof|Claim:]] ''' that the definition above actually defines an [[equivalence relation]]
{{Begin Inline Proof}}{{:Equivalence of Cauchy sequences/Proof}}
{{End Proof}}{{End Theorem}}
==See also==
* [[Completeness]]
* [[Complete metric space]]
* [[Completion of a metric space]]
* [[Equivalence relation]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Metric Space|Topology|Functional Analysis|Analysis|Real Analysis}}
[[Category:Equivalence relations]][[Category:Exemplary pages]][[Category:First-year friendly]]