inequality - Convergence of $L^p$ norms

Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that
$\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, f\in L^q\cap L^\infty, \textrm{ then } \lim_{p \to \infty}\|f\|_p = \|f\|_\infty$ (whih by the way justify this notation)

This convergence implies the following:
$\forall f, \forall \epsilon > 0, \exists q:= q(f,\epsilon),\textrm{ such that } \forall p\geq q, |\|f\|_p-\|f\|_\infty| < \epsilon$

This means that given an approximation error of the infinity norm bounded by $\epsilon$, I should be able to compute an (let's call it) index so that, I don't need to go any further, but I have a priori knowledge of the potential error.

The idea is, I am working on some pattern recognition problems and I am using the infinity norm somewhere there. However, as it is quite unreliable against outliers, using a p-norm approximation allows to "average out" the local outliers and get a more robust result. The higher the power the more importance the local outliers (or singularities) have, and the less I like it :)

If you have any idea on a proof or a results, it would be very helpful.