Lebesgue integration

Question

if $f : \mathbb{R} \to \mathbb{R}$ is continuous function which is Lebesgue integrable on $\mathbb{R}$ then show that there is sequence $x_n$ which goes to infinity and $x_n f(x_n)$ goes to $0$. Since $f$ is Lebesgue integrable on $\mathbb{R}$. It is finite almost every where. Also $f$ is continuous what can we use it to prove our result. Please, provide any hint.

Answer 1

Suppose not. Then there exists $\epsilon>0$ and $A>0$ so that $$|xf(x)|\ge\epsilon\quad(x>A).$$ Hence $\int_A^\infty|f(x)|\,dx\ge\epsilon\int_A^\infty\frac{dx}x=\infty$.