Understanding an equality

Question

I need to understand why this is true because it is a given to a proof on why f(x)= $\sqrt x$ continuous on [0,∞).

|√x −√y| ≤ |√x + √y|

Thanks

Answer 1

We know that $\sqrt{x} \geq 0 \; \forall x \geq 0$. So $$ \sqrt{x}-\sqrt{y} \leq \sqrt {x} \leq \sqrt {x} + \sqrt {y} \\ = |\sqrt {x} + \sqrt {y}|$$ and $$\sqrt{y}-\sqrt{x} \leq \sqrt {y} \leq \sqrt {y} + \sqrt {x} \\ = |\sqrt {y} + \sqrt {x}|$$ Therefore, $$|\sqrt {x}-\sqrt {y}| \leq |\sqrt {x}+\sqrt {y}|.$$