How to show this integral limit identity?

Let I be a generalized rectangle and let $f: I to mathbb{R}$. Show that $$lim_{ptoinfty}left(int_I|f|^pright)^{1/p} = max|f|.$$ I found it straightforward to show that these integrals in the limit sequence are properly defined and I showed the LHS $leq$ RHS by using the definition of the integral using partitions. I am confused about the approach to showing that $RHS leq LHS$ to give equality. Thanks for any hints and suggestions.