Given two non-negative numbers $a$ and $b$, I'm trying to found an upper bound for $-\sqrt{a} + \sqrt{a+b}$
But I'd like the bound not to be dependable on any square roots and have only one term. Namely, I want something like $O(a+b)$. Anyone knows how to obtain such a bound?
$$\begin{align} \sqrt{a+b}-\sqrt{a} &=\left(\sqrt{a+b}-\sqrt{a}\right)\frac{\sqrt{a+b}+\sqrt{a}}{\sqrt{a+b}+\sqrt{a}}\\ &=\frac{a+b-a}{\sqrt{a+b}+\sqrt{a}}\\ &=\frac{b}{\sqrt{a+b}+\sqrt{a}}\\ &\le\frac{b}{\sqrt{a+b}}\\ &\le\frac{b}{\sqrt{b}}\\ &=\sqrt{b}\\ &\le1+b.~\blacksquare \end{align}$$