Do Cauchy Sequences Imply Convergent Differences?

In summary: It seems to prove the statement false.In summary, the conversation discusses the use of properties of absolute value and the triangle inequality to prove a statement involving Cauchy sequences. Different examples are given to test the validity of the statement, but there is uncertainty about the strictness of the inequality and the choice of the constant K. Further discussion is needed to clarify these points.
  • #1
Bptrhp
8
4
Homework Statement
Let ##(x_n)## and ##(y_n)## be Cauchy sequences in ##\mathbb{R}## such as ##x_n-y_n\rightarrow 0##. Prove that if exists ##K>0## such as ##|x_n|\leq K,\forall \,n\in\mathbb{N}##, then there exists ##n_0\in\mathbb{N}## such as ##|y_n|\leq K, \forall \,n>n_0## .
Relevant Equations
##|x_n|\leq K,\forall \,n\in\mathbb{N}##
I've started by writing down the definitions, so we have

$$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$
$$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w $$
$$(y_n) \,\text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|y_m-y_n|<w $$

I tried using properties of the absolute value and the only vaguely useful result I got is ##|x_n-t_n|\leq C+|t_n|##. I can't see how to use this to prove the desired result.
Any hints? I appreciate any help!
 
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  • #2
What about ##x_n=K## for all n, ##y_n = K+1/n##? It seems to prove the statement false. Is the inequality supposed to be strict?
 
  • #3
What if ##x_n=1=K## and ##y_n=1+\dfrac{1}{n}##?
 
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  • #4
Office_Shredder said:
What about ##x_n=K## for all n, ##y_n = K+1/n##? It seems to prove the statement false. Is the inequality supposed to be strict?

fresh_42 said:
What if ##x_n=1=K## and ##y_n=1+\dfrac{1}{n}##?

Your example is not a counter-example: The quantifier in the premise is existential, so if [itex]K = 1[/itex] doesn't work you should take a larger value of [itex]K[/itex]. In this case [itex]K \geq 2[/itex] and [itex]n_0 = 1[/itex] works.

@Bptrhp: You have a bound on [itex]|x_n|[/itex] and you need to find a bound on [itex]|y_n|[/itex]. So use the triangle inequality in the form [tex]
\begin{align*}
|y_n| &= |y_n - x_n + x_n| \\ &\leq |y_n - x_n| + |x_n|. \end{align*}[/tex]
 
  • #5
pasmith said:
Your example is not a counter-example: The quantifier in the premise is existential, so if [itex]K = 1[/itex] doesn't work you should take a larger value of [itex]K[/itex]. In this case [itex]K \geq 2[/itex] and [itex]n_0 = 1[/itex] works.

@Bptrhp: You have a bound on [itex]|x_n|[/itex] and you need to find a bound on [itex]|y_n|[/itex]. So use the triangle inequality in the form [tex]
\begin{align*}
|y_n| &= |y_n - x_n + x_n| \\ &\leq |y_n - x_n| + |x_n|. \end{align*}[/tex]
You don't get to pick K. It says if there exists ##K## such that ##K\geq |x_n|## then stuff about it is true. We gave an example of such a K, so the stuff about it should be true.
 
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  • #6
pasmith said:
Your example is not a counter-example: The quantifier in the premise is existential, so if [itex]K = 1[/itex] doesn't work you should take a larger value of [itex]K[/itex]. In this case [itex]K \geq 2[/itex] and [itex]n_0 = 1[/itex] works.
I don't see that at all.
 
  • #7
Office_Shredder said:
What about ##x_n=K## for all n, ##y_n = K+1/n##? It seems to prove the statement false. Is the inequality supposed to be strict?
Even with strict inequality we have: ##x_n = 1 - \frac 1 n## and ##y_n = 1 + \frac 1 n##.
 
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Related to Do Cauchy Sequences Imply Convergent Differences?

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. This means that for any small value, there exists a point in the sequence after which all subsequent terms are within that value of each other.

2. How is a Cauchy sequence different from a convergent sequence?

A Cauchy sequence is a type of convergent sequence, but not all convergent sequences are Cauchy sequences. The main difference is that a convergent sequence has a specific limit to which all terms in the sequence tend to, while a Cauchy sequence only requires the terms to become arbitrarily close to each other.

3. What is the importance of Cauchy sequences in mathematics?

Cauchy sequences are important in mathematics because they provide a rigorous definition for the concept of convergence. They are used in many areas of mathematics, including analysis, calculus, and number theory.

4. How do you prove that a sequence is a Cauchy sequence?

To prove that a sequence is a Cauchy sequence, you must show that for any small value, there exists a point in the sequence after which all subsequent terms are within that value of each other. This can be done by using the definition of a Cauchy sequence and the properties of the terms in the sequence.

5. Can a Cauchy sequence have more than one limit?

No, a Cauchy sequence can only have one limit. This is because the definition of a Cauchy sequence requires the terms to become arbitrarily close to each other, and if there were multiple limits, the terms would not be getting closer to each other. If a Cauchy sequence has more than one limit, it is not a Cauchy sequence.

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