A few years ago I picked up the book *An Introduction to Analysis* (by James Kirkwood) at a textbook clearance sale. I finally got around to studying it recently during my commute time. For me, it’s mostly stuff that was touched on in college courses but I’ve never fully approached from the analysis angle. (I give this introduction so it doesn’t just look like I’m asking for help with homework.)

I’m in the very first section, where it’s setting up the real numbers. The problem I have a question about is this :

Prove that |a| <= |b| if and only if a[sup]2[/sup] <= b[sup]2[/sup].

Now, given what’s been in the text so far, the only way I can figure doing this is to consider every case and what results – though effective, it seems a tad inelegant. What really makes me think there is a better way is the ‘hint’ given at the back of the book :

|a| = sqrt( a[sup]2[/sup] )

Now, I’m not sure that proving that is any easier, but what I’m wondering is, where does that get me? The very next problem is to prove that x[sup]s[/sup] < y[sup]s[/sup] for positive x,y and s rational, so I’m wondering if this is supposed to be similar or not.

Anyone see what the book is getting at?

It’s difficult to give everything in the text to this point, but I’ll give the statements I have regarding order and absolute value:

F is an ordered field if :

There is a nonempty subset P of F which is closed under addition & multiplication.

For any a in F exactly one holds : a is in P, a = 0, or -a is in P.

The ordering operation :

For an ordered field F, and P as described, for a,b in F:

a < b if b - a is in P.

The absolute value :

For real a, |a| is defined as :

|a| = a, if a >= 0; -a, if a < 0.