NL-NWO 1968 Problem 2

It holds: $N,a > 0$ . Prove that $\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}$ , and if $N \ge 1$ and $a = [\sqrt{N}]$ . Prove that if $a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)$ is a better approximation for $\sqrt{N}$ than $a$ .