MHB An Equivalence Relation with Cauchy Sequences

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The discussion centers on defining an equivalence relation on the set of Cauchy sequences in \mathbb{Q} using the limit condition |x_n - y_n|. Participants explore whether subsequences are necessary for proving reflexivity, symmetry, and transitivity, concluding that the triangle inequality suffices. One user initially questions the need for subsequences but later realizes the hint provided was misplaced. The consensus is that subsequences are not required for this proof. Overall, the focus remains on the properties of the equivalence relation without the need for subsequences.
Aryth1
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We let [math]C[/math] be the set of Cauchy sequences in [math]\mathbb{Q}[/math] and define a relation [math]\sim[/math] on C by [math](x_i) \sim (y_i)[/math] if and only if [math]\lim_{n\to \infty}|x_n - y_n| = 0[/math]. Show that [math]\sim[/math] 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 [math]|x-y|\geq 0[/math] for all [math]x,y\in \mathbb{Q}[/math].

Any help is appreciated!

EDIT: Wasn't less than, but greater than.
 
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Aryth said:
We let [math]C[/math] be the set of Cauchy sequences in [math]\mathbb{Q}[/math] and define a relation [math]\sim[/math] on C by [math](x_i) \sim (y_i)[/math] if and only if [math]\lim_{n\to \infty}|x_n - y_n| = 0[/math]. Show that [math]\sim[/math] 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 [math]|x-y|\geq 0[/math] for all [math]x,y\in \mathbb{Q}[/math].

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.
 
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!
 
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