Why $\frac{|f(x)-f(x_0)|}{|\sqrt{f(x)}+\sqrt{f(x_0)}|}\leq |f(x)-f(x_0)|$ is considered true in an SE answer?

Question

I was reading the last answer by Wyatt Kuehster in this Stack Exchange question.

There he uses the inequality:

$$\frac{|f(x)-f(x_0)|}{|\sqrt{f(x)}+\sqrt{f(x_0)}|}\leq |f(x)-f(x_0)|$$

It seems to me that this inequality is incorrect as the denominator can be less than 1.

For eg: let $f(x)=0.09$ and $f(x_0)=0.04$. Then denominator $=0.5$.

Then why is the inequality used in that answer? The answer got two upvotes. Or am I missing anything?