Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function such that the sequence \( f(x), f(2x), f(3x), \dots \) converges to \( 0 \) for any \( x > 0 \). Prove or disprove that \[ \lim_{x \to \infty} f(x) = 0. \]