Suppose that $f,u,u_n\in L^p(\Omega)$ where $\Omega\subset\mathbb{R}^N$ is a bounded domain and $f,u,u_n\geq 0$. Suppose $\|u+u_n\|_p\rightarrow \|u+f\|_p,$ $u_n\rightarrow f\text{ a.e. in } \Omega,$ and $u_n>f.$
Does this implies that $\|u_n-f\|_p\rightarrow0\,?$
Thanks.