Q:

Product of weak and strong convergent sequences in $L^p$

I already saw some proofs here with $b_n\to b$ in $L^2$ and $a_n\rightharpoonup a$ in $L^2$. Then
$$
\int a_n b_n \to \int a b.
$$
But what goes wrong if both sequences are weak convergent?
Proof:
$$
\int_\Omega a_nb_n\,dx \to \int_\Omega ab\,dx.
$$ 
Indeed,
$$
a_nb_n-ab=a_n(b_n-b)+(a_n-a)b
$$
Then
$$
\Big|\int_\Omega a_n(b_n-b)\,dx\,\Big|\le 
\|a_n\|_{L^2}\|b_n-b\|_{L^2}\le M\|b_n-b\|_{L^2}\to 0,
$$
and
$$
\int_\Omega (a_n-a)b \to 0, 
$$
as $a_n-a\rightharpoonup 0$.
Why does not hold
$$ \int_\Omega a_n(b_n-b)\,dx\,\to 0,$$
if $b_n\rightharpoonup b$.

A:

Then it may fail. Let $a_n(x)=b_n(x)=\sin(n\,x)$. By the Riemamm-Lebesgue lemma, $a_n$ and $b_n$ converge weakly to $0$ on $[-\pi,\pi]$, but
$$
 a_n\,b_n=\sin^2(x)=\frac{1-\cos(2\,n\,x)}{2}
$$
converges wekly to $1/2$.

