If $a,b\geq 0$ show that $\left| \sqrt{a}-\sqrt{b}\right|\leq\sqrt{\left|a-b\right|}$.
WLOG we can assume that $a\geq b$. If one of them is $0$ this is trivial. So assume none of them is $0$. Now, $$\left| \sqrt{a}-\sqrt{b}\right|\leq\sqrt{\left|a-b\right|} \iff a-2\sqrt{ab}+b\leq a-b \iff -2\sqrt{ab}\leq -2b \iff \sqrt{ab}\geq b \iff ab\geq b^2\iff a\geq b$$
Is this proof correct?
