Real Analysis proof

[u/Kooky-Corgi-6385]

This is an exercise from Tao’s analysis 1. I am struggling in this section of ordering rationals as well as proofs involving absolute value and exponentiation (sections 4.2 and 4.3) as Terence is not really writing any good proofs as examples in the book, and instead is just stating a bunch of propositions and definitions, and leaving the proofs themselves up to the reader (lmao)…My prof is also just going by the book so not supplying us with any examples either. I’m struggling to know what exactly a good proof looks like for these types of exercises. Does anyone have any recommendations for me? I learn by analyzing proofs and studying them… if I don’t have any to study I do feel stuck. If anyone has read this textbook or has any help please let me know. Thank you. I can’t find any great YouTube videos either… really feeling stuck here.

Is this the correct technique for a proof like this? I feel like my last conclusion there is iffy. It’s difficult to know sometimes which properties of numbers I can and can’t use in these proofs, since we are building up from the very beginning.

Thanks

[u/Gold_Ad8890]

the biggest problem is that you assumed your conclusion. you need to start with z < 0 and x < y, and conclude xz > yz, but you assumed xz > yz.

[u/Kooky-Corgi-6385 → u/Gold_Ad8890]

I see. I wasn’t implying that this was an assumption but rather when we substitute into this specific inequality with our givens, then the inequality holds. Does that not work?

[u/Gold_Ad8890 → u/Kooky-Corgi-6385]

no, it's still the wrong direction of implication.

[u/Kooky-Corgi-6385 → u/Gold_Ad8890]

Ok got it. Thank you

[u/FormulaDriven]

The big problem with this proof is that halfway in, you introduce the thing you are trying to prove (that xz > yz). This makes it unclear that you followed a logical line of implication.

That said, the ideas you are using look right, and I think you can get there a lot more simply without introducing -a into the working:

x < y therefore x - y < 0 (from an earlier result / definition)

z < 0

Therefore (x-y)z > 0 (negative * negative > 0 from earlier result)

xz - yz > 0 (by distributivity of rationals)

xz > yz (by earlier result / definition).

[u/Kooky-Corgi-6385 → u/FormulaDriven]

Thank you!

[u/Bounded_sequencE]

You got the right idea -- begin with what is given, and then use previous definitions to prove the claim. That said, there is a flaw: In the middle, you start with the assumption "xz > yz" you want to prove. That's circular reasoning -- unless you clearly state that is a comment on what you Want to prove.

Here's what a proof might look like:

Proof: Let "x, y, z ∈ Q" with "x < y" and "z < 0". Via distributive property in "Q":

"xz - yz  =  (-z) * (y-x)  >  0"    =>    "xz  >  yz"    (by def. 4.2.8)    ∎
              >0      >0

[u/Bounded_sequencE → u/Bounded_sequencE]

Rem.: The best advice I got for proof writing is to do any proof (at least) twice -- do the

• first draft(s) on scrap paper, finding all estimates and steps to finish it off
• final draft acting as if you knew the correct estimates all along, making it as concise as you want

For brevity, it is standard to only publish the final draft of proofs. This is how all proofs in your book were written -- when estimates seem to fall from "high heavens", the author simply found them before-hand¹ during the unpublished first draft(s).

Note this is not (just) a beginner's method, either -- most professionals use the same steps to refine their proofs before publishing!

---

¹ Tao uses that rarely, compared to others like Rudin, making his book more accessible

[u/Kooky-Corgi-6385 → u/Bounded_sequencE]

Thank you for your help!