TITLE: Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?
QUESTION [6 upvotes]: Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?
Because of the nature of the square root function, its derivative monotonically decreases. so differences "further up" the function would be less than those lower down.

REPLY [6 votes]: Yes.
$$
\left|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}\right| =\frac{\lvert x^2-y^2\rvert}{\left|\sqrt {a^2+x^2} + \sqrt {a^2+y^2}\right|}
= |\sqrt {x^2} - \sqrt {y^2}|\cdot \frac{|\sqrt {x^2} + \sqrt {y^2}|}{\left|\sqrt {a^2+x^2} + \sqrt {a^2+y^2}\right|}
\leq |\sqrt {x^2} - \sqrt {y^2}|
$$

REPLY [4 votes]: This is true. You can see this by assuming $x>y$ without losing generality and then differentiating
\begin{equation}
f(a)=\sqrt{a^2+x^2}-\sqrt{a^2+y^2}
\end{equation}
with respect to $a$. Derivative is negative hence $f$ is decreasing function of $a$ and is maximized at $0$.