Monotone conitnuous function - find limits

[daniel_i_l]

Homework Statement

if f is monotonic and continues in R, and $$\int^{\infty}_{a} f(x)dx$$ converges then
$$lim_{x \rightarrow \infty} xf(x) = 0$$

Homework Equations

The Attempt at a Solution

I know that if xf(x) converges at all then it has to convergs to 0. But how do I know that it converges?
Thanks.

 

[Dick]

Can you show that for any positive value of k however small, that f(x)<k/x for all x>N for N sufficiently large? Hint: suppose it's not.

 

[daniel_i_l]

Thanks.