# 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 !