Q:

Prove this $\sup f_n$ has infinite integral

Let $f_n\in L^1[0,1]$ be positive functions such that:
$$\int_0^1 f_n=1$$
$$\int_{\frac{1}{n}}^{1}f_n<\frac{1}{n}$$
for all n, prove:
$$\int_0^1 \sup_{n\geq 1}f_n =\infty$$
I don't know how to handle this, who can give me some suggestions? 
Thank all of you! 

A:

Hint: If $\int_0^1 g<\infty$, then $\int_0^{\frac1n}g\to0$ as $n\to\infty$.

