Let $X$ be a measure space and let $f_{n}$ be a sequence of functions which converge pointwise to a function $f$ in $L^{p}(X)$ where $p>1$ and suppose $g_{n}$ is a sequence of functions which converge pointwise to a function $g$ in $L^{\frac{p}{p-1}}(X)$. Prove that:
$$ \lim_{n \to \infty} \int_{X} f_{n}(x) g_{n}(x) dx = \int_{X} f(x)g(x) dx.$$
No idea how to proceed. Any help? I guess somewhere we need Hölder and DCT.
Not true: Let $f_n(x)$ be the a constant times the characteristic function of $[1/n,2/n]$ and $g_n(x)$ be another constant times the characteristic function of $[1/n,2/n]$, the constants defined that the $L^p$ and $L^{p \over p-1}$ norms of $f_n$ and $g_n$ respectively are both 1. Then they both converge pointwise to zero, so that the right hand side is zero. But the left integral is always 1. Thus what you're saying isn't even true if $f_n$ and $g_n$ are in $L^p$ and $L^{p \over p-1}$ respectively.