Difference between revisions of "Equivalence of Cauchy sequences/Definition"
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==Definition== | ==Definition== | ||
</noinclude>Given two [[Cauchy sequence|Cauchy sequences]], {{M|1=(a_n)_{n=1}^\infty}} and {{M|1=(b_n)_{n=1}^\infty}} in a [[metric space]] {{M|(X,d)}} we define them as [[equivalence relation|equivalent]] if{{rAPIKM}}: | </noinclude>Given two [[Cauchy sequence|Cauchy sequences]], {{M|1=(a_n)_{n=1}^\infty}} and {{M|1=(b_n)_{n=1}^\infty}} in a [[metric space]] {{M|(X,d)}} we define them as [[equivalence relation|equivalent]] if{{rAPIKM}}: | ||
| − | * {{MM|1=\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(a_n,b_n)<\epsilon]}} | + | * {{MM|1=\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(a_n,b_n)<\epsilon]}}<noinclude> |
| − | <noinclude> | + | |
==References== | ==References== | ||
<references/> | <references/> | ||
Latest revision as of 20:12, 29 February 2016
This sub-page is ideal for transclusion, where ever a reminder of the definition of equivalence of Cauchy sequences is required.
Definition
Given two Cauchy sequences, [ilmath](a_n)_{n=1}^\infty[/ilmath] and [ilmath](b_n)_{n=1}^\infty[/ilmath] in a metric space [ilmath](X,d)[/ilmath] we define them as equivalent if[1]:
- [math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(a_n,b_n)<\epsilon][/math]
