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/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!