Question regarding convergence in the pth mean Here's what I am trying to prove:
Let $\Omega = [0,1]$, $1<p <\infty$. Let $\{ f_n \}$ be a sequence in $L^p [0,1]$ such that $f_n \to f$ almost everywhere and $f \in L^p [0,1]$. Suppose that there is some $M \in \mathbb R$ such that $\lVert f_n \rVert _p \le M$ for all $n$. Prove that for $g\in L^q [0,1]$ where $1/p + 1/q =1$, we have $\lim \int f_n g  d\lambda = \int fg d\lambda$. Here $\lambda$ is the Lebesgue measure.

Here's my poor attempt:
Let $\{ f_n  \}$ be a sequence of functions in $L_p [0,1]$. Since $\lVert \cdot \rVert _p$ is a continuous function on $L^p [0,1]$ and $\lVert f_n \rVert _p \le M$ for all $n$, we have that $\lVert f \rVert _p \le M$.
If we try to estimate $\lvert \int f_n g d\lambda - \int fg d\lambda \rvert \le \lVert f_n -f \rVert _p \lVert g \rVert _q$. If we could somehow how that $\lVert f_n - f \rVert _p \to 0$ as $n \to \infty$, we will be done.
There are certain things that I observe: we have a finite measure space and so almost everywhere convergence implies convergence in measure. However, convergence in measure will not possibly imply convergence in the $p$th mean. So we are hopeless at this certain point.  However, I notice that I am not using the fact that $\lVert f_n \rVert \le M$ and $\lVert f \rVert \le M$ for each $n \in \mathbb N$. I do not see how to use it as well.
I am looking for hints that could possibly lead me to a solution to this problem. Any series of hints will be appreciated.

You simply have
$$\left|\int f_ng-\int fg\right|\leq \int|f_n-f||g|\underset{\text{Holder ineq.}}{\leq} \|f_n-f\|_p\|g\|_q\underset{n\to \infty }{\longrightarrow }0.$$
The claim follow.
