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## Homework Statement

Prove that ||a|-|b||[tex]\leq[/tex] |a-b| for all a,b in the reals

## Homework Equations

I know we have to use the triangle inequality, which states:

|a+b|[tex]\leq[/tex] |a|+|b|.

Also, we proved in another problem that |b|[tex]\leq[/tex] a iff -a[tex]\leq[/tex]b[tex]\leq[/tex]a

## The Attempt at a Solution

Using the other problem we solved, we can say -|a-b| [tex]\leq[/tex] ||a|-|b||[tex]\leq[/tex] |a-b|

then I said...

|b|=|(b-a)+a|

then I use the Triangle Inequality"

|(b-a)+a|[tex]\leq[/tex] |b-a| + |a|

|b-a| + |a| [tex]\leftrightarrow[/tex] |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 !