Rigorous proof of basic "absolute value" theorem?

[scorpion990]

Hey :) I'm working through a Real Analysis text, and I came across this theorem and "proof":
http://img352.imageshack.us/img352/6725/proofbx2.png

It kind of took me by surprise, because the author of the text is usually very careful about defining every little operation, and took 2-3 pages to show that you can actually "add a constant to both sides" of an equation and equality without changing the solution set.

Except for the sloppy notation, two parts of the "proof" confused me. First, why can you add "plus or minus a" and "plus or minus b" to get "plus or minus (a+b)"? I'm also confused as to how the author jumped from plus or minus (a+b) to |a+b|.

Maybe I'm just slow. Any thoughts?

[quasar987]

It's not so much that ±a + ±b = ±(a+b) as much as the fact that if ±a<=c and ±b<=d, then a+b<=c+d and -a-b<=c+d; that is to say, ±(a+b) <= c+d.

As to why ±(a+b)<=|a|+|b| implies |a+b|<=|a|+|b|, it follows from the definition of the absolute value that states that |x|=x if x>=0 and -x if x<0. Here, if (a+b)>=0, then (a+b)=|a+b| and we have |a+b|= (a+b) <=|a|+|b|. In the other case, (a+b)<0, we have -(a+b)=|a+b|, and here again, |a+b|=-(a+b)<=|a|+|b|.

[scorpion990]

Oh.. I see.
The author did prove the necessary theorems for me to understand why:
\pm(a+b)\leq|a|+|b| \foralla,b\inR.

Now, if I understand this correctly, this can really be broken down into 5 cases:
1. +(a+b)<|a| + |b|, where a+b is positive.
2. -(a+b)<|a| + |b|, where a+b is positive.
3. +(a+b)<|a| + |b|, where a+b is negative.
4. -(a+b)<|a| + |b|, where a+b is negative.
5. (a+b)<|a| + |b|, where a+b is 0.

For |a+b| < |a| + |b| to be true, you stated that the only three conditions that need to be true are:
1. +(a+b)<|a| + |b|, where a+b is positive.
2. -(a+b)<|a| + |b|, where a+b is negative.
3. (a+b)<|a| + |b|, where a+b is 0.

...which are all true. So, in a way, |a+b| < |a| + |b| is just a special case of a more general theorem. Is my logic here correct?