RO-NMO 2003 Problem 113

Let be two functions $f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R}$ having that properties that $f$ is continuous, $g$ is nondecreasing and unbounded, and for any sequence of rational numbers $\left( x_n \right)_{n\ge 1}$ that diverges to $\infty ,$ we have $$1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) .$$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) .$ Radu Gologan