Module Inequalities.

Question

I have to prove:

$$|a| -| b| \leq |a-b|$$

Using the known result:

$$|a+b| \leq |a|+|b|, $$

$a, b \in \mathbb{R}.$

What I did;

Used the known result and simple manipulation, i got to;

$$|a|-|b| \leq 2 \cdot |a|-|a+b|$$

I am not sure how to show

$$ 2 \cdot |a|-|a+b| =|a-b|$$

Answer 1

You have $$ |b|=|a+(b-a)|\leq|a|+|b-a|. $$ So $$\tag{1} |b|-|a|\leq|b-a|.$$ As the roles of $a$ and $b$ are symmetric, you also get $$\tag{2}|a|-|b|\leq|b-a|.$$ Combining $(1)$ and $(2)$, $$ \left|\,|a|-|b|\,\right|\leq|b-a|. $$