I want to check that $\left|\left|a+b\right|-\left|a\right|-\left|b\right|\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $.

Question

I want to check that $\left|\left|a+b\right|-\left|a\right|-\left|b\right|\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $.

It 's equivalent to $\left|a+b\right|-\left|a\right|-\left|b\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $ and $-2\left|b\right|\leq\left|a+b\right|-\left|a\right|-\left|b\right|\forall a,b\in\mathbb{R} $.

The first statement: $\left|a+b\right|-\left|a\right|-\left|b\right|\leq0 $ so we are done.

The second statement is equivalent to $-\left|b\right|\leq\left|a+b\right|-\left|a\right|\forall a,b\in\mathbb{R}\Longleftrightarrow\left|a\right|-\left|b\right|\leq\left|a+b\right|\forall a,b\in\mathbb{R} $

Do I have mistakes?

Thanks a lot.

Answer 1

Here is a start. We have

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

$$ \implies | a+b| \leq |a| + |b|, \\ -(|a|+|b|) \leq -|a|+|b| $$

Adding the above inequalities gives

$$ | a+b|-(|a|+|b|) \leq 2|b| . $$

Now, try to find the other inequality.

Answer 2

For any $a,b\in\mathbb R$, the following inequality holds. $$|a|-|b|\le ||a|-|b||\le |a+b|.$$ The latter equality holds when $ab\le0.$ Hence, your first inequality is true.