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Recently in class we have learned how to create the number systems and have just defined the definitions of rational Null sequences and Cauchy sequences but with $- 1/k \leq x_i \leq 1/k$ terminology instead of $\epsilon$ and the use of absolute value.

For many of the proofs such as the sum of two null sequences being null, I’d like to use the triangle inequality as it is what I know from real analysis.

As I am working with the rationals, I can’t assume what we know about the Real numbers yet and so is it safe for me to use $|x_i|\leq 1/k$ when doing the proofs or is that incorrect terminology? It would make things so much easier.

Andrés E. Caicedo
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Partey5
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  • You can safely use the triangle inequality. If you know some inequality holds for all reals, then it holds for all rationals. Granted, you'd need to re prove it for the case of rational numbers, since the reals "don't exist" yet, but usually the same proof goes through. Things break down between $\mathbb{Q}$ and $\mathbb{R}$ when a $\sup$ is involved. – Reveillark Nov 17 '19 at 22:27
  • The question is a bit vague, but I think the point is that you are "creating" real numbers in the class, so you cannot use them yet.

    However, things like $1/k$ do make sense, because they are rational numbers, which come from quotients of integers.

    And inequalities also make sense with rational numbers, because they translate into inequalities between integers, and these again translate to whether some integer is positive.

     – WhatsUp Nov 17 '19 at 22:30
  • If I wanted to prove every rational null sequence is a rational Cauchy sequence, the triangle inequality is the only method I know how to do. Is there another way without using triangle inequalities? – Partey5 Nov 17 '19 at 22:34

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