What about this example:

$\displaystyle f_n:=\frac{1}{n*x}$

$\displaystyle \forall x \in (0,\frac{1}{2}) \rightarrow |f_n(x)|\ge \frac{1}{n*\frac{1}{2*n}}=2$

and $\displaystyle \int_{[0,1]} |f_n|^p = \frac{1}{n^p * (1-p)} $

I think this one clearly works now, yes?