IMC 2005 · Problem 5

Let $f : (0,\infty) \to \mathbb{R}$ be a twice continuously differentiable function such that

$$|f''(x) + 2xf'(x) + (x^2+1)f(x)| \le 1$$

for all $x$. Prove that $\lim\limits_{x \to \infty} f(x) = 0$.