real analysis - Showing $int_E f=lim_{ntoinfty}int_E f_n$ for all measurable $E$

The following is an exercise from Carothers' Real Analysis:

Suppose $f$ and $f_n$ are nonnegative, measurable functions, that $f=\lim_{n\to\infty} f_n$ and that $\int f=\lim_{n\to\infty}\int f_n<\infty$. Prove that $\int_E f=\lim_{n\to\infty}\int_E f_n$ for any measurable set $E$. (Hint: Consider both $\int_E f$ and $\int_{E^c} f$.) Give an example showing that this need not be true if $\int f=\lim_{n\to\infty}\int f_n=\infty$.

Attempt:

Using Fatou's Lemma, I can get that

$$\int_E f=\int_E \lim_{n\to\infty} f_n=\int_E\liminf_{n\to\infty} f_n\leq \liminf_{n\to\infty}\int_E f_n=\lim_{n\to\infty}\int_E f_n$$

However, I'm doubtful that this is correct since the hint in the book said to consider both $E^c$ and $E$. I know that if $E$ is measurable, then its complement is measurable as well. A small push in the right direction would be appreciated. Thanks.

Answer

Consider the following:

$$\int f = \int_E f + \int_{E^c} f \le \varliminf \int_E f_n + \varliminf \int_{E^c} f_n \le \varliminf\left(\int_E f_n + \int_{E^c} f_n\right) = \varliminf \int f_n = \int f$$