What does the N mean in a Cauchy sequence definition?

[Scott4775]

What does the "N" mean in a Cauchy sequence definition?

Hi everyone,

I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question.

I believe I have an intuitive understanding of what a Cauchy sequence means: given any ε>0 the difference between the terms in a sequence approaches zero as the index of the terms (n,m) go to infinity. In other words, as one looks at larger and larger terms, the terms get closer and closer together. (Please pardon that I'm not fluent in Latex).

Many of the definitions say something about "...given an ε>0, there exists an N such that..."

What does the "N" mean. What is its significance in the definition of a Cauchy sequence?

I hope this was explained well enough and I appreciate all help!

Thanks!

 

[Dick]

Hi everyone,

I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question.

I believe I have an intuitive understanding of what a Cauchy sequence means: given any ε>0 the difference between the terms in a sequence approaches zero as the index of the terms (n,m) go to infinity. In other words, as one looks at larger and larger terms, the terms get closer and closer together. (Please pardon that I'm not fluent in Latex).

Many of the definitions say something about "...given an ε>0, there exists an N such that..."

What does the "N" mean. What is its significance in the definition of a Cauchy sequence?

I hope this was explained well enough and I appreciate all help!

Thanks!

You summed that up pretty well. As you said "get closer and closer together", ε is how close they are. And as you said "one looks at larger and larger (n,m)", N is how large they have to be to be that close. The formal definition just sums up exactly what you said in words.

 

[lurflurf]

N which may depend upon epsilon is the cutoff beyond which terms are near. All but some finite number of the infinite points in the sequence are near (|xn-xm|<epsilon). If n and m are both>N then |xn-xm|<epsilon, if not this may not hold.

 

[Scott4775]

Thank you both for your responses. Both of you have cleared up any confusion that I had. I really appreciate it. I searched online but couldn't quite find the answer I was looking for in plain language. :)

Thanks again!