If x and y are both greater than or equal to 1, show that $|\sqrt{x}-\sqrt{y}|$ is less than or equal to $0.5| x-y |$

Question

If x and y are both greater than or equal to 1, show that $|\sqrt{x}-\sqrt{y}|$ is less than or equal to $0.5| x-y |$

Would really appreciate any help! Thanks

Answer 1

Just write, with $x\geq1, \, y \geq1$: $$|\sqrt{x}-\sqrt{y}|=\left|\frac{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}{\sqrt{x}+\sqrt{y}}\right|=\left|\frac{x-y}{\sqrt{x}+\sqrt{y}}\right|\leq\left|\frac{x-y}{1+1}\right|=\frac{\left|x-y\right|}{2}.$$

Answer 2

Replacing $x$ by $x^2$ and $y$ by $y^2$ all reduces to

$$2\le|x+y|$$ that is true with $x\ge1$ and $y\ge1$.