Real Analysis Absolute values

Question

Someone please help me with detailed explanation on how to solve this problem.

For all $a, b \in \Bbb R$, show that; $$ | a - b | \geq | a | - | b | $$

Answer 1

Using the triangle inequality:

$$|a|=|a-b+b|\leq |a-b|+|b|$$

Thus $|a|-|b|\leq |a-b|$.

Similarly you can prove that $|b|-|a|\leq |a-b|$

So we have $\left||a|-|b|\right|\leq |a-b|$ which is the reverse triangle inequality.