**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.**
##Inequalities
...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
**Proof**
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 . **(!)**
##Absolute values
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
**Theorem**
**Proof (by cases)**
1. ,
If , then it follows
2.
then , and again,
3.
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 proofnotbycases.*
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.
**Proposition: **
**Proof**
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.
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