An Equivalence Relation with Cauchy Sequences

In summary: Just out of curiosity, is there a way to prove that this is an equivalence relation using subsequences? I'm not sure why the hint was presented to us in the first place.EDIT: I figured out that the hint provided for this problem was meant for another problem. It was an incorrect placement. Thanks for your help, Opalg!
  • #1
Aryth1
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We let \(\displaystyle C\) be the set of Cauchy sequences in \(\displaystyle \mathbb{Q}\) and define a relation \(\displaystyle \sim\) on C by \(\displaystyle (x_i) \sim (y_i)\) if and only if \(\displaystyle \lim_{n\to \infty}|x_n - y_n| = 0\). Show that \(\displaystyle \sim\) is an equivalence relation on C.

We were given a hint to use subsequences, but I don't think they are really necessary... Are they?

I don't need help with the proof, per say, I would just like an opinion of whether or not you think subsequences are necessary.

Reflexivity and Symmetry were easy to show without the use of subsequences, and transitivity seems to follow from the triangle inequality and the fact that \(\displaystyle |x-y|\geq 0\) for all \(\displaystyle x,y\in \mathbb{Q}\).

Any help is appreciated!

EDIT: Wasn't less than, but greater than.
 
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  • #2
Aryth said:
We let \(\displaystyle C\) be the set of Cauchy sequences in \(\displaystyle \mathbb{Q}\) and define a relation \(\displaystyle \sim\) on C by \(\displaystyle (x_i) \sim (y_i)\) if and only if \(\displaystyle \lim_{n\to \infty}|x_n - y_n| = 0\). Show that \(\displaystyle \sim\) is an equivalence relation on C.

We were given a hint to use subsequences, but I don't think they are really necessary... Are they?

I don't need help with the proof, per say, I would just like an opinion of whether or not you think subsequences are necessary.

Reflexivity and Symmetry were easy to show without the use of subsequences, and transitivity seems to follow from the triangle inequality and the fact that \(\displaystyle |x-y|\geq 0\) for all \(\displaystyle x,y\in \mathbb{Q}\).

Any help is appreciated!

EDIT: Wasn't less than, but greater than.
Hi Aryth, and welcome to MHB!

I agree with you, the triangle inequality is all you need here.
 
  • #3
Opalg said:
Hi Aryth, and welcome to MHB!

I agree with you, the triangle inequality is all you need here.

Thank you!

Just out of curiosity, is there a way to prove that this is an equivalence relation using subsequences? I'm not sure why the hint was presented to us in the first place.

EDIT: I figured out that the hint provided for this problem was meant for another problem. It was an incorrect placement. Thanks for your help, Opalg!
 
Last edited:

FAQ: An Equivalence Relation with Cauchy Sequences

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between two elements in a set. It is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity. In simpler terms, it is a way of classifying elements in a set into groups based on certain criteria.

2. What are Cauchy sequences?

Cauchy sequences are sequences of real numbers that converge to a limit. They are named after the French mathematician Augustin-Louis Cauchy and are used to define the completeness of a metric space. In simpler terms, a Cauchy sequence is a sequence in which the terms get closer and closer together as the sequence progresses.

3. How is an equivalence relation related to Cauchy sequences?

An equivalence relation with Cauchy sequences is a specific type of equivalence relation that is defined on the set of Cauchy sequences. It is used to classify Cauchy sequences into equivalence classes, where two sequences are considered equivalent if they converge to the same limit. This relation is important in the study of real analysis and metric spaces.

4. What are the three properties of an equivalence relation with Cauchy sequences?

The three properties of an equivalence relation with Cauchy sequences are: reflexivity, symmetry, and transitivity. Reflexivity means that every Cauchy sequence is equivalent to itself. Symmetry means that if two sequences are equivalent, then their reverse sequences are also equivalent. Transitivity means that if two sequences are equivalent and one of them is also equivalent to a third sequence, then the first and third sequences are also equivalent.

5. Why are equivalence relations with Cauchy sequences important in mathematics?

Equivalence relations with Cauchy sequences are important in mathematics because they help us classify sequences into groups and understand the behavior of these sequences. They are also essential in the study of real analysis and metric spaces, as they are used to define concepts such as completeness and convergence. Furthermore, they have applications in other areas of mathematics, such as topology and abstract algebra.

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