Fatou's lemma and $\limsup \int_C |f_n|^p d\mu$

Question

Let $f_1,f_2,... ,f \in \mathscr{L}^p(X,\mathscr{A},\mu)$, where ${f_n}\to f$ a.e. and $\lim_n ||f_n||_p=||f||_p$.

We can find $A \in \mathscr{A}$ s.t. ${f_n}\to f$ uniformly with $\mu(A^c)< \epsilon$, and $\lim_n \int_A |f_n-f|^p d\mu=0$.

Let $C=X-A$. Is it possible to show $\limsup_n \int_C |f_n|^p d\mu \leq \int_C |f|^p d\mu$ with the help of Fatou's lemma?