Problem: Suppose that $f:\Re^+\to\Re^+$ is a continuous function with the following property: for all $x\in\Re^+$, the sequence $f(x), f(2x),f(3x)\cdots$ tends to $0$. Prove that $\lim\limits_{t\to\infty}f(t) = 0.$

Please help to solve this problem, teacher, said act to the contrary, but I do not understand how there podstupitsya.Znayu that is somehow connected with the theorem of Baire.This is a homework assignment.

Thanks!