for "a" and "b" are real numbers prove that:
ImageShack - Image Hosting :: 34860604zw7.gif
for "a" and "b" are real numbers prove that:
ImageShack - Image Hosting :: 34860604zw7.gif
$\displaystyle \left| a \right| \leqslant \left| {a - b + b} \right| \leqslant \left| {a - b} \right| + \left| b \right|\; \Rightarrow \;\left| a \right| - \left| b \right| \leqslant \left| {a - b} \right|$
$\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} $
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
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 $\displaystyle
\left| a \right| \leqslant \left| b \right|\text{ if and only if } - \left| b \right| \leqslant a \leqslant \left| b \right|$.