Let $1 \leq p \leq q \leq +\infty$ real such as $1/p+1/q=1$. We have $f_n, f \in L^p(\mathbb{R})$ and $g_n, g \in L^q(\mathbb{R})$ such as $f_n \rightarrow f$ in $L^p$ and $g_n \rightarrow g$ in $L^q$. I would like to prove that $f_n g_n \rightarrow fg$ in $L^1$.
I wrote that $f_n g_n \in L^1$ and $||f_n g_n||_1 \leq ||f_n||_p ||g_n||_q$ by Hölder inequality. Similarly, $fg \in L^1$ and $||fg||_1 \leq ||f||_p ||g||_q$.
$||f_n||_p \rightarrow ||f||_p$ and $||g_n||_q \rightarrow ||g||_q$ but despite all these elements, I can't prove that $||f_n g_n - fg||_1 \rightarrow 0$. Someone could help me ? Thank you in advance.