hi, I stumbled upon a simple demonstration but i can´t solve it:
||z|-|w|| <= |z-w|
Can someone help me?
Pedro
This proof relies on this fact: $\displaystyle a \geqslant 0\;\& \, - a \leqslant b \leqslant a\, \Leftrightarrow \,\left| b \right| \leqslant \left| a \right|$.
So $\displaystyle \left| u \right| \leqslant \left| {u - w} \right| + \left| w \right|\, \Rightarrow \,\left| u \right| - \left| w \right| \leqslant \left| {u - w} \right|$
Likewise $\displaystyle \left| w \right| - \left| u \right| \leqslant \left| {w - u} \right| = \left| {u - w} \right|$
Thus we now have $\displaystyle - \left( {\left| {u - w} \right|} \right) \leqslant \left( {\left| u \right| - \left| w \right|} \right) \leqslant \left( {\left| {u - w} \right|} \right) $
Can you use the fact to finish?
Please start a new thread for any new question.
You said that you understand that $\displaystyle a \geqslant 0\;\& \, - a \leqslant b \leqslant a\, \Leftrightarrow \,\left| b \right| \leqslant \left| a \right| .$
If you do, can we say that $\displaystyle |a|\le ||a||~?$
If so then is it not true that $\displaystyle -|a|\le a \le |a|~?$
The fact is, that any nonnegative number is equal to its absolute value. So if $\displaystyle \displaystyle a \geq 0$, then $\displaystyle \displaystyle a = |a|$.
However, any negative number is less than its absolute value (since the absolute value is always nonnegative). So if $\displaystyle \displaystyle a < 0$, then $\displaystyle \displaystyle a < |a|$.
So that means for any $\displaystyle \displaystyle a$ that $\displaystyle \displaystyle a \leq |a|$.
A similar argument works the other way to show $\displaystyle \displaystyle -|a| \leq a$.
So that means $\displaystyle \displaystyle -|a| \leq a \leq |a|$.