Hi, I'm having difficulty understanding this definition.

A sequence

of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N

|xm-xn|< ε

It seems simple, but the point I don't understand is N. What is it? What is the function of it?
I'll be grateful if you could explain this to me. Thanks in advance!

2. Originally Posted by truevein
Hi, I'm having difficulty understanding this definition.

A sequence

of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N

|xm-xn|< ε

It seems simple, but the point I don't understand is N. What is it? What is the function of it?
I'll be grateful if you could explain this to me. Thanks in advance!
N is a natural number, not a function. It is just some number that you pick. The terms of the sequence are enumerated by natural numbers. So they picked the N-th term and say, "for everyone after that, this will happen..."

Example, lets say you pick N = 3, then the statement refers to the terms: $\displaystyle \displaystyle x_4,~x_5,~x_6, \dots$ ($\displaystyle \displaystyle x_n \text{ and }x_m$ are just two of these terms, any two you want to pick)

If N = 126, then the statment refers to: $\displaystyle \displaystyle x_{127}, ~x_{128}, \dots$

So basically the definition is saying: A sequence is Cauchy if after a certain point (marked by the N-th term), all the terms of the sequence are really close together (that is, within a distance of $\displaystyle \displaystyle \varepsilon$ of each other), no matter which 2 you pick. It would basically mean that the terms are convergng to some point, and so, they will all get very close to each other as they all get very close to said point. You will learn (or should have learned) that a sequence converges if and only if it is Cauchy. That is a good context to think of it in.

Any questions?

3. Originally Posted by Jhevon
N is a natural number, not a function. It is just some number that you pick. The terms of the sequence are enumerated by natural numbers. So they picked the N-th term and say, "for everyone after that, this will happen..."

Example, lets say you pick N = 3, then the statement refers to the terms: $\displaystyle \displaystyle x_4,~x_5,~x_6, \dots$ ($\displaystyle \displaystyle x_n \text{ and }x_m$ are just two of these terms, any two you want to pick)

If N = 126, then the statment refers to: $\displaystyle \displaystyle x_{127}, ~x_{128}, \dots$

So basically the definition is saying: A sequence is Cauchy if after a certain point (marked by the N-th term), all the terms of the sequence are really close together (that is, within a distance of $\displaystyle \displaystyle \varepsilon$ of each other), no matter which 2 you pick. It would basically mean that the terms are convergng to some point, and so, they will all get very close to each other as they all get very close to said point. You will learn (or should have learned) that a sequence converges if and only if it is Cauchy. That is a good context to think of it in.

Any questions?
Thanks for the answer, I think I understand now..The N does not matter then, since the sequence is infinite. But what is the difference between an ordinary convergent sequence and a Cauchy sequence?
The existence of N? Thanks for your precious time

4. Originally Posted by truevein
Thanks for the answer, I think I understand now..The N does not matter then, since the sequence is infinite. But what is the difference between an ordinary convergent sequence and a Cauchy sequence?
The existence of N? Thanks for your precious time
every converging sequence is Cauchy sequence, and every Cauchy sequence is limited

5. Originally Posted by truevein
Thanks for the answer, I think I understand now..The N does not matter then, since the sequence is infinite. But what is the difference between an ordinary convergent sequence and a Cauchy sequence?
The existence of N? Thanks for your precious time
Oh no, the N matters (and will vary depending on what sequence you are dealing with and what $\displaystyle \varepsilon$ you are given). For a Cauchy sequence, you have to be be able to pick a point for which the definition is true. The fact that you can do that is what makes the sequence Cauchy. You can't do that for the sequence $\displaystyle \displaystyle \{ x_n \} = n$ for example. It matters that no matter what N you pick, the terms after that don't get closer together. Successive terms are always 1 unit apart, and if you pick terms that are 1000 terms apart, the distance between them would not be small, it would be a thousand--which, chances are, won't be smaller than any arbitrary $\displaystyle \varepsilon$ we are given. The fact that you can choose some N and all the terms after it do what you say, that is, become really close together, is what makes the sequence Cauchy. And there is no difference between a Cauchy sequence and a convergent one. Convergence implies Cauchy and Cauchy implies convergence. Saying a sequence is Cauchy is logically equivalent to saying it is convergent.

6. Originally Posted by Jhevon
Oh no, the N matters (and will vary depending on what sequence you are dealing with and what $\displaystyle \varepsilon$ you are given). For a Cauchy sequence, you have to be be able to pick a point for which the definition is true. The fact that you can do that is what makes the sequence Cauchy. You can't do that for the sequence $\displaystyle \displaystyle \{ x_n \} = n$ for example. It matters that no matter what N you pick, the terms after that don't get closer together. Successive terms are always 1 unit apart, and if you pick terms that are 1000 terms apart, the distance between them would not be small, it would be a thousand--which, chances are, won't be smaller than any arbitrary $\displaystyle \varepsilon$ we are given. The fact that you can choose some N and all the terms after it do what you say, that is, become really close together, is what makes the sequence Cauchy. And there is no difference between a Cauchy sequence and a convergent one. Convergence implies Cauchy and Cauchy implies convergence. Saying a sequence is Cauchy is logically equivalent to saying it is convergent.
Oh, sorry for being such a dumb. Now I understand the N point matters and is the basic rule of Cauchy sequences. I can't thank you enough for explaining this!

7. you should know that :

real number $\displaystyle a \in \mathbb{R}$ is limit of the sequence $\displaystyle x_n$ if for every $\displaystyle \varepsilon >0$ there is natural number $\displaystyle n_0 \in \mathbb{N}$ such that for every natural number $\displaystyle n\ge n_0$ implies $\displaystyle |x_n-a|<\varepsilon$

$\displaystyle (\forall \varepsilon >0) (\exists n_0 \in \mathbb{N}) (\forall n\in \mathbb{N}) (n\ge n_0 \Rightarrow |x_n -a | < \varepsilon )$

and that means

$\displaystyle \displaystyle \lim_{n\to \infty} x_n = a$

if that is true than sequence $\displaystyle x_n$ converges to $\displaystyle a$

or you can look at it like this

let the $\displaystyle (X,d)$ is metric space. for sequence $\displaystyle (x_n)_{n\in \mathbb{N}} \subset X$ we say that is Cauchy sequence if

$\displaystyle (\forall \varepsilon >0) (\exists n_0=n_0(\varepsilon) \in \mathbb{N}) (\forall n,m \in \mathbb{N}) (n,m\ge n_0 \Rightarrow d(x_n,x_m) < \varepsilon )$

meaning that, sequence is Cauchy sequence if

$\displaystyle \displaystyle \lim _{n,m \to \infty } d(x_n,x_m) = 0$

8. Originally Posted by yeKciM
you should know that :

real number $\displaystyle a \in \mathbb{R}$ is limit of the sequence $\displaystyle x_n$ if for every $\displaystyle \varepsilon >0$ there is natural number $\displaystyle n_0 \in \mathbb{N}$ such that for every natural number $\displaystyle n\ge n_0$ implies $\displaystyle |x_n-a|<\varepsilon$

$\displaystyle (\forall \varepsilon >0) (\exists n_0 \in \mathbb{N}) (\forall n\in \mathbb{N}) (n\ge n_0 \Rightarrow |x_n -a | < \varepsilon )$

and that means

$\displaystyle \displaystyle \lim_{n\to \infty} x_n = a$

if that is true than sequence $\displaystyle x_n$ converges to $\displaystyle a$

or you can look at it like this

let the $\displaystyle (X,d)$ is metric space. for sequence $\displaystyle (x_n)_{n\in \mathbb{N}} \subset X$ we say that is Cauchy sequence if

$\displaystyle (\forall \varepsilon >0) (\exists n_0=n_0(\varepsilon) \in \mathbb{N}) (\forall n,m \in \mathbb{N}) (n,m\ge n_0 \Rightarrow d(x_n,x_m) < \varepsilon )$

meaning that, sequence is Cauchy sequence if

$\displaystyle \displaystyle \lim _{n,m \to \infty } d(x_n,x_m) = 0$
Thank you, I think we studied this in school:P but just the N part had confused me a lot.

9. Originally Posted by truevein
Thank you, I think we studied this in school:P but just the N part had confused me a lot.
heheheh... sorry I misunderstand you

perhaps it's better to index some natural number after which all members of sequence are in region of some point with $\displaystyle n_0$ rather than with N which is used to associate numbers in set of natural numbers

10. That's true, it took long for me to understand which N it is when I first saw it

11. Originally Posted by yeKciM
heheheh... sorry I misunderstand you

perhaps it's better to index some natural number after which all members of sequence are in region of some point with $\displaystyle n_0$ rather than with N which is used to associate numbers in set of natural numbers
Originally Posted by truevein
That's true, it took long for me to understand which N it is when I first saw it
Perhaps. You can use whatever suits your fancy, it is a matter of taste. in the end, they are just dummy variables. There is a difference between upper case n (N) and the bold-faced n used to denote the naturals ($\displaystyle \displaystyle \mathbb N$). Even when writing on paper you write them differently. in general, there is no cause for confusing the two. but there is no law for using something other than "N" here, unless it is used for something else...

12. Originally Posted by Jhevon
Perhaps. You can use whatever suits your fancy, it is a matter of taste. in the end, they are just dummy variables. There is a difference between upper case n (N) and the bold-faced n used to denote the naturals ($\displaystyle \displaystyle \mathbb N$). Even when writing on paper you write them differently. in general, there is no cause for confusing the two. but there is no law for using something other than "N" here, unless it is used for something else...
yes i agree but one my not know how to write different those two $\displaystyle N$ and $\displaystyle \mathbb{N}$
in any case it's just index of the member in sequence nothing more