Upper bound on a difference of two square root functions

Question

I am trying to find an upper bound on $\sqrt{x+y}-\sqrt{x}$, $x\geq 0,y\geq -x$.

I would think $\sqrt{x+y}-\sqrt{x}\leq \sqrt{|y|}$ but I'm not sure how to prove that because although both sides are positive, squaring them does not prove the desired inequality. Any suggestions?

Answer 1

It is not true both sides are positive (take $x> 0$ and $y =-x$). Anyway, $\sqrt{x+y}-\sqrt{x}\leq \sqrt{|y|}$ is equivalent to $$\sqrt{x+y}\leq \sqrt{x} + \sqrt{|y|} \tag {1}.$$ Here both sides are $\ge 0$, thus $(1)$ is equivalent to $$x + y \le x + 2\sqrt{x}\sqrt{|y|} + |y| .$$ This is obviously true.