TITLE: From $C[0, 1]$ to $L^\infty [0,1]$
QUESTION [5 upvotes]: I am dealing with the following exercise:
Let $u_n$ bounded in $L^\infty[0,1]$ such that, for any continuous function $f: [0,1]\times R$ to $R$
$$\lim_n \int_0^1 f(x, u_n(x))=\int_0^1 f(x, u(x)).$$
Prove that $u_n$ converges to $u$ in $L^1.$
There is a hint saying to prove it first for $u\in C^0$.
I succeeded in proving it for $u$ continuous, choosing $f(x, y)=|y- u(x)|$, but I don't know how to get to $L^\infty$.
My idea was to approximate $u$ in $L^1$ with continuous functions, (while continuous functions are not dense in $L^\infty$, right?), but I can't see how to work out details. 
Thank you in advance.

REPLY [4 votes]: On a finite measure space $(X,\mu)$, $L^q(X) \subset L^p(X)$ for $q\geq p$. This is proved by Holder's inequality. Apply this to be able to approximate your $L^\infty$ functions.