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}