Prove a monotone increasing function is less than another function

Question

The question asks to prove $\sqrt{x} < x$ for all $x>1$ This is for a Foundations of Analysis class.

Answer 1

$\sqrt{x}\sqrt{x}>1\sqrt{x}$ iff $(\sqrt{x}-1)\sqrt{x}>0$.

Since $\sqrt{x}>1$, this is true by closure of the positive reals.