# More Cauchy equivalence

Question:

Prove that if a Cauchy sequence $x_1, x_2,...$ of rationals is modified by changing a finite number of terms, the result is an equivalent Cauchy sequence.

All the math classes I have taken previously were computational, and my textbook contains almost no definitions.

So, I know that the properties of reflexivity, symmetry, and transitivity must be present to have an equivalence. However, I don't know how to show that Cauchy sequences are equivalent. I'm guessing that if they converge to the same limit, then they are equivalent, but that's just my guess. The book remains mute on this and many points.

Once I know the criteria for equivalence (whatever that may be), my next problem is that I don't have a clue how to start this or any proof. I think I need to have a way to represent the sequences.
Lets call them: A, B (modified A), and C (modified B). I think I need three objects to show transitivity.
Now here's the sticky part; I don't know how to represent A,B, or C. My brain has not yet made the jump into abstraction. Any suggestions on a way to write A,B, and C would be greatly appreciated!

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HallsofIvy
Homework Helper
You can't do it without knowing what "equivalence" means in this particular case!

I suspect that two sequences are equivalent if they converge to the same limit, but you really should check your definitions.

shmoe
Homework Helper
Notation suggestion- use $$x_1,x_2,\ldots$$ as your first sequence, $$x_1,x_2,\ldots$$ as your modified sequence.

Two Cauchy sequences are equivalent if the sequence of their differences, $$x_1-y_1,x_2-y_2,\ldots$$ converges to zero. They are eventually getting "close" to one another. This isn't quite the same as asking the two sequences to have the same limit point, but they are trying to converge to the same thing (the rationals are not complete, so they may not converge at all within the rationals).

This should be in your book though, and I'm making an assumption on the context (it looks like you're heading towards constructing the reals from the rationals via Cauchy sequences.

Tom Mattson
Staff Emeritus
Two Cauchy sequences are equivalent if the sequence of their differences, $$x_1-y_1,x_2-y_2,\ldots$$ converges to zero.