Suppose $\int_{1}^\infty f(x)dx$ converges, and$ \frac{f(x)}{x}$ is monotone decreasing on $[1,\infty]$.
How can I prove $\lim_{x \to \infty}xf(x)= 0$ ?
Suppose $\int_{1}^\infty f(x)dx$ converges, and$ \frac{f(x)}{x}$ is monotone decreasing on $[1,\infty]$.
How can I prove $\lim_{x \to \infty}xf(x)= 0$ ?