Show that if $f : E \rightarrow [0,\infty]$, $\lim \limits _{k\rightarrow \infty} f_k = f$ on $E$, and $f_k \leq f$ on $E$ for each $k \in N$, then $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k = \int \limits _E f$
An idea was to show that $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k \leq \int \limits _E f$ and $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k \geq \int \limits _E f$. I am able to prove $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k \leq \int \limits _E f$ but am struggling to prove the second condition. My idea was to use fatou's lemma to get $ \int \limits _E \liminf \limits _{k\rightarrow \infty} f_k = \int \limits _E \lim \limits _{k\rightarrow \infty} f_k = \int \limits _E f \leq \liminf \limits _{k\rightarrow \infty} \int \limits _E f_k = \lim \limits _{k\rightarrow \infty} \int \limits _E f_k $ but I don't know how to show $\liminf \limits _{k\rightarrow \infty} f_k = \lim \limits _{k\rightarrow \infty} f_k$, besides showing $\liminf \limits _{k\rightarrow \infty} f_k = \limsup \limits _{k\rightarrow \infty} f_k$ .
Any ideas on how I could finish this? Also, is my approach wrong? Could I do it a better way?