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# Criterion of Cauchy

I seem to have some trouble with understanding this criterion fully...

 Cauchy wrote: Let (an) be a sequence [R or C]. We say that (an) is a Cauchy sequence if, for all ε > 0 there exists N ∈ N such that m, >= N =⇒ |am − an| < ε.

So what I think that I understand:
No matter how small I take my epsilon, I will always find a smaller number (if the series is convergent) by subtracting two numbers of a row.

For example:
Row (goes to the square root of 2): 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, ...
Let's take epsilon = 1/100.000
Then I can always find a number smaller than epsilon:
1.414213 - 1.41421 = 3*10^-6 < epsilon

And if epsilon is even smaller, you just go further down the row to get 2 numbers that are even closer to each other. It makes sense to me because if you go really far in the row, the distance between two numbers is getting smaller and smaller until it reaches 0 at infinity which means there is a number and therefor convergent.

Questions:

• Are my thoughts (semi-)correct?
• What is N? I don't see it in the actual formula?
• Can someone fill in the formula with my example (so what's am, an, m, n and N in the formula?) and work it out, because on every website I visited they just take a row and in one line they say it is convergent without a numerical solution.

Afaceinthematrix
 Cauchy wrote: Let (an) be a sequence [R or C]. We say that (an) is a Cauchy sequence if, for all ε > 0 there exists n ∈ N such that m, >= N =⇒ |am − an| < ε.

I would change that first "N" to an "n" so that it makes more sense.

Question 1:
Your thoughts are semi-correct and you have the right idea. Intuitively, a Cauchy sequence is a sequence in which the length between each term gets smaller and smaller. For instance, the sequence an = 1/n is Cauchy. 1/1 - 1/2 = 1/2. 1/2 - 1/3 = 1/6... Etc. In complete metric spaces, Cauchy sequences converge. So the sequence 1/n converges (although the harmonic series diverges!).

To find the exact n, it isn't always going to be a chug and plug formula. You just want to think about it. It would be harder in your example than in my example. All the n is saying that if you choose any epsilon (and this works for epsilon arbitrarily small and greater than 0), the difference will be less than that epsilon for EVERY term after n. For instance, in my example, choose ε = 1/5.

Then we have the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6....

1 - 1/2 = 1/2 !< 1/5
But, 1/2 - 1/3 = 1/6 < 1/5

So am - an = 1/2 - 1/3...

And so our "n" is 2, because if you choose any m greater than n - 1/3 for instance - then the difference is less than our ε. I hope this helps. I don't really want to try to find n in your example because it would be a pain in the ass and this should explain the concept...
saberlivre
If we choose ε = 0.000005 (in the series of the square root of 2), we would see that for any m > 6 we have |am - a6 | < 0.000005, which shows that it is a Cauchy sequence.

rjraaz
I liked this post cauchy criterion is also not understandable to me,

@Afaceinthematrix: Looks like something was lost in the definition while copy pasting:

 Afaceinthematrix wrote: Let (an) be a sequence [R or C]. We say that (an) is a Cauchy sequence if, for all ε > 0 there exists n ∈ N such that m, >= N =⇒ |am − an| < ε.

Before copy paste:
 Quote: Let (an) be a sequence [R or C]. We say that (an) is a Cauchy sequence if, for all ε > 0 there exists N ∈ IN (natural numbers, 1,2,3,4, etc) such that m, n >= N ⇒ |am − an| < ε.

Anyway, I think I get it now, so kudos to you guys.