Elementary Analysis, Triangle Inequality Help

[Gooolati]

Homework Statement

Prove that ||a|-|b||$$\leq$$ |a-b| for all a,b in the reals

Homework Equations

I know we have to use the triangle inequality, which states:
|a+b|$$\leq$$ |a|+|b|.

Also, we proved in another problem that |b|$$\leq$$ a iff -a$$\leq$$b$$\leq$$a

The Attempt at a Solution

Using the other problem we solved, we can say -|a-b| $$\leq$$ ||a|-|b||$$\leq$$ |a-b|

then I said...
|b|=|(b-a)+a|
then I use the Triangle Inequality"
|(b-a)+a|$$\leq$$ |b-a| + |a|

|b-a| + |a| $$\leftrightarrow$$ |a-b| + |a|

and someone said this proved the first side of the inequality, but I have no idea why.

All help is appreciated ! Thanks !

 

[murmillo]

I think it will be hard to use the triangle inequality, because whenever you use a minus sign, the expression will always have to be in parentheses. In other words, I don't think the triangle inequality will help you go from, say, |a-b| to something where there's a minus sign outside parentheses. I was able to solve the problem by proving the inequality for each possible case, namely
Case 1: a and b are positive
Case 2: a and b are negative, a>b
Case 3: a and b are negative, a<b
Case 4: a is positive and b is negative. (You don't need to have case 5 where b is positive and a is negative, since both cases turn out to be the same.)
The triangle inequality was not necessary when I did the problem.