An Equivalence Relation with Cauchy Sequences

[Aryth1]

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.

 

[Opalg]

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.

 

[Aryth1]

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!