**Note: I lost a page from my notepad, so this is not a continuous stream of notes in that if we let ... Okay now I'm trailing off into a math joke, anyway.**
...along with definitions of , , , and ,
Everything else comes from Propositions *[for inequalities]*. Firstly, we know using the order axioms (10 - 12) that for any two and , one and only one of the following hold,
So suppose and , then
or in shorthand, .
Another useful assertion: and , then
If , , so . By similar reasoning, . So by P12,
and we conclude, . There is a special consequence if , for
In fact, since , , we have the useful result: , so . **(!)**
Given if , we define the **absolute value**,
Noting that is always positive, except when , which is neither positive nor negative. For example, .
##The Triangle Inequality
**Proof (by cases)**
If , then it follows
then , and again,
We must prove that . So suppose , then we need to prove
Or in other words, . But this is clearly true, since .
Otherwise, if , we need to prove that
or more simply, . But this is also true, since we assumed , so .
The final case is achieved by interchanging and in the final proof.
**Proof (a bit shorter, and more slick)**
If we denote as the strictly positive square root of , then
is an equivalent definition of the absolute value, then
Taking the square root of both sides, we conclude since we know that , however the proof of this is left as exercise.
*As a sidenote, this proof reminds me of the total unnecessary use of proof by cases. Fuck proof by cases, someone please prove that if there exists a proof by cases, greater abstraction allows for a proof-not-by-cases.*
Closer examination finds , when the parity of and are similar (or if one of and is zero). If and have opposite parity, then it will be the case that
##Further notes on
We proved that , by establishing , and dividing through by . Why, though, do we assume ? This cannot be established with all axioms up the P12, which we now proceed to resolve.
We have seen that , so it follows immediately from order axioms that , and thus by definition, in particular, .
#Final notes on basic properties of (real) numbers
- Seems absurd and trite, but let's remember that it's difficult to justify, rigorously, (since we appeal to familiarity), the statements of P1 - P12.
- We work with numbers all the time, however just *what* numbers are, remains philosophically and precisely vague.
- We do know, however, that regardless of what numbers are (at least the ones we are used to dealing with, *cough* , *cough*), that they obey axioms 1 through to 12.
##A further note during studying
After finishing studiously writing these notes on a notepad, I flipped through the next questions in Spivak's Preface, and promptly had scholastic, intellectual heart attack.
A cautious look at the next chapter resulted in an exponential increase in the intensity of said heart attack. Yeah, this should be fun.
I should stick to the ANU Analysis I course notes...they're easier.