Rigorous proof of basic "absolute value" theorem? 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?