Why is the last step of my proof for convergence in L^{p} space correct?

[SqueeSpleen]

If $f_{n} \underset{n \to \infty}{\longrightarrow} f$ in $L^{p}$, $1 \leq p < \infty$, $g_{n} \underset{n \to \infty}{\longrightarrow} g$ pointwise and $|| g_{m} ||_{\infty} \leq M \forall n \in \mathbb{N}$ prove that:
$f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} fg$ in $L^{p}$

My attemp:

$\displaystyle \int | f_{n} g_{n} - f g_{n} |^{p} = \displaystyle \int | g_{n} | | f_{n} - f |^{p} \leq M^{p} \displaystyle \int | f_{n} - f |^{p}$
Then $f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} f g_{n}$ in $L^{p}$
Now let's prove that $f g_{n} \underset{n \to \infty}{\longrightarrow} fg$ in $L^{p}$
$g_{n} \longrightarrow g$ a.e. $\Longrightarrow g_{n} \underset{\longrightarrow}{m} g$
$\forall \varepsilon > 0, \forall \delta > 0, \exists n_{0} \in \mathbb{N} / \forall n \geq n_{0} :$
$| D | = | \{ x / | g_{n} (x) - g(x) | \geq \delta \} | < \varepsilon$

$\displaystyle \int | f g_{n} - fg |^{p} = \displaystyle \int | f |^{p} | g_{n} - g |^{p} \leq \displaystyle \int_{D} | f |^{p} M^{p} + \displaystyle \int_{D^{c}} | f |^{p} \delta^{p}$
I know $| D | < \varepsilon$, but $f$ isn't necessarily bounded in $D$, I need to prove that $\int_{D} | f |^{p} \longrightarrow 0$ as $| D | \to 0$

Anyone has any idea?
Is this approach right or the last step is false and I need to rework the proof enterely?

Edit 2: Edit 1 was completely wrong so I deleted it.

 

[dirk_mec1]

You'll need to redo the proof start by adding an additional term and immediately subtracting it then use your bounds and the fact that gn => g and fn=>f pointwise.

 

[SqueeSpleen]

You mean to be able to apply dominated convergence theorem?

Anyway, I asked my teacher and he knew filled the last gap in my proof.

As
$\displaystyle \int | f |^{p} < \infty \Longrightarrow \displaystyle \int_{E} | f |^{p} < \infty \forall E \subset R^{n}$
Then by the absolute continuity of the Lebesgue integral, for every $\varepsilon > 0$ there exists a $\delta ' > 0$ such that:
$| E | < \delta 0 \Longrightarrow | \displaystyle \int_{E} | f |^{p} | < \varepsilon$
As $(2M)^{p}$* is a constant, this converges to zero as $\delta \to 0$ and $\varepsilon \to 0$
* I made a little mistake in the previous post:
$| g - g_{n} |^{p} \leq (2M)^{p}$ not only $M^{p}$