How to prove the following inequality: $a,b,c\in\mathbb{R},|\sqrt{a^2+b^2}-\sqrt{a^2+c^2}|\leq|b-c|$?

Question

How to prove the following inequality? $$a,b,c\in \mathbb{R},|\sqrt{a^2+b^2}-\sqrt{a^2+c^2}|\leq|b-c|$$

The absolute values really confused me, I tried to square both sides but it doesn't seem to work?

Answer 1

\begin{align*} \bigg|\|(a,b)\|_{2}-\|(a,c)\|_{2}\bigg|\leq\|(a,b)-(a,c)\|_{2}=\|(0,b-c)\|_{2}=|b-c|. \end{align*}

Answer 2

Alternatively, using just the triangle inequality and some elementary algebra: $\left|\sqrt{a^2+b^2} - \sqrt{a^2+c^2}\right|=\dfrac{|b^2-c^2|}{\sqrt{a^2+b^2}+\sqrt{a^2+c^2}}\le \dfrac{|b-c||b+c|}{|b|+|c|}\le |b-c|$.