General Lebesgue Dominated Convergence Theorem In Royden (4th edition), it says one can prove the General Lebesgue Dominated Convergence Theorem by simply replacing $g-f_n$ and $g+f_n$ with $g_n-f_n$ and $g_n+f_n$.  I proceeded to do this, but I feel like the proof is incorrect.
So here is the statement:

Let $\{f_n\}_{n=1}^\infty$ be a sequence of measurable functions on $E$ that converge pointwise a.e. on $E$ to $f$.  Suppose there is a sequence $\{g_n\}$ of integrable functions on $E$ that converge pointwise a.e. on $E$ to $g$ such that $|f_n| \leq g_n$ for all $n \in \mathbb{N}$.  If $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $g_n$ = $\int_E$ $g$, then $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $f_n$ = $\int_E$ $f$.

Proof:
$$\int_E (g-f) =  \liminf \int_E g_n-f_n.$$
By the linearity of the integral: 
$$\int_E g - \int_E f = \int_E g-f \leq \liminf \int_E g_n -f_n = \int_E g - \liminf \int_E f_n.$$
So,
$$\limsup \int_E f_n \leq \int_E f.$$
Similarly for the other one.
Am I missing a step or is it really a simple case of replacing.

I ignore the E and proceed to prove it as required by Exercise 2.20 in Folland. 
The problem is essentially a Generalized Dominated Convergence Theorem which I prove by reworking the proof of Lebesgue's Dominated Convergence Theorem introduced in class. 
$|f_n| \le g_n$ implies $-g_n \le f_n \le g_n$ which implies $f_n + g_n \ge 0$ and $g_n-f_n \ge 0$
We know $ \lim (f_n + g_n) = f + g $ and $ \lim (g_n - f_n) = g -f$. We can therefore apply Fatou's Lemma to the above cases to get: 
(I) $\int g + \int f = \int (g+f) = \int  \lim (f_n + g_n) = \int \liminf (f_n + g_n) \le \liminf \int (f_n + g_n) = \liminf \int g_n + \liminf \int f_n = \int g + \liminf \int f_n$
Thus, $\int f \le \liminf \int f_n$.
(II) $\int g - \int f = \int (g-f) = \int  \lim (g_n - f_n) = \int \liminf (g_n - f_n) \le \liminf \int (g_n - f_n) = \liminf \int g_n + \liminf \int - f_n = \int g - \limsup \int f_n$
Thus,  $\int f \ge \limsup \int f_n$.
(III) Notice $\{ \int f_n \}_{n \in \mathbb{N}}$ is fundamentally a sequence. So, as is true for any sequence: $$\liminf \int f_n \le \limsup \int f_n$$
(IV) Also, our work with Fatou's has established the following: $$\limsup \int f_n \le \int f \le \liminf \int f_n$$
Combining (III) and (IV), we get $\liminf \int f_n = \limsup \int f_n = \int f$ . The first equality implies the existence of $\lim \int f_n$ and the second establishes: $$\lim \int f_n = \int f = \int \lim f.$$
