Let $x>0$ be some variable that I am interested in analyzing in the limit $x\to\infty$. Let $y = x+c$ for some constant $c$.
Is $\sqrt{y} - \sqrt{x} \geq f(c) > 0$ for some function $f$ of only $c$?
This feels "obvious", but I am having quite a bit of difficulty in establishing it, so perhaps I am just Very Wrong.