1. ## a proof question..

for "a" and "b" are real numbers prove that:
ImageShack - Image Hosting :: 34860604zw7.gif

2. $\displaystyle \left| a \right| \leqslant \left| {a - b + b} \right| \leqslant \left| {a - b} \right| + \left| b \right|\; \Rightarrow \;\left| a \right| - \left| b \right| \leqslant \left| {a - b} \right|$
$\displaystyle \begin{gathered} \left| b \right| \leqslant \left| {b - a + a} \right| \leqslant \left| {b - a} \right| + \left| a \right|\; \Rightarrow \;\left| b \right| - \left| a \right| \leqslant \left| {b - a} \right| = \left| {a - b} \right| \hfill \\ - \left| {a - b} \right| \leqslant \left| a \right| - \left| b \right| \leqslant \left| {a - b} \right| \hfill \\ \end{gathered}$

3. ## how??

first we have
|a-b|>=||a|-|b||

what did you do at the first step in order to transform it into

http://img152.imageshack.us/img152/5374/52524050qr2.gif

4. That is a simple application of the triangle inequality.
In fact, the entire problem depends upon the triangle inequality.

It also depends upon the general fact $\displaystyle \left| a \right| \leqslant \left| b \right|\text{ if and only if } - \left| b \right| \leqslant a \leqslant \left| b \right|$.

5. can you tell me
what operation you did in each step of this prove

??