SE-SKOLORNAS 1966 Problem 5

Let $f(r)$ be the number of lattice points inside the circle radius $r$ , center the origin. Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$ , put $g(r) = f(r) - kr^2$ . Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$ ?