Let $ X=[0,1] $, $ \mu $ the Lebesgue measure on $ X $ and $ \{ f_n \}_{n=1}^{\infty} $ a sequence of functions in $ \mathcal{L}^p(\mu) $, $ 1 < p < \infty $. Suppose that there exists an $ f \in \mathcal{L}^p(\mu) $ such that $ f_n \rightarrow f $ pointwise on $ X $. Also, suppose that there is a positive constant $ M $ such that $ || f ||_p \le M \: \: \forall $ $ n \in \mathbb{N} $. Then prove that for each $ g \in \mathcal{L}^q(\mu) $, we have $$ \lim_{n \rightarrow \infty} \int_{X} (f_ng) d\mu = \int_{X} (fg) d\mu $$ where $ q $ is the conjugate exponent of $ p $.
What I tried: By Holder's inequality, $$ \left| \int_{X} (f_ng) d\mu - \int_{X} (fg) d\mu \right| \le \int_{X} |f_n-f||g| d\mu \le ||f_n-f||_p||g||_q$$ Therefore, to prove the assertion, it suffices to prove that $ ||f_n-f||_p \rightarrow 0 $, i.e $ f_n \rightarrow f $ in $ \mathcal{L}^p(\mu) $
Let $ \epsilon > 0 $. Note that $ |f|^p \in \mathcal{L}^1(\mu) $, so by absolute continuity, there exists a $ \delta > 0 $ such that $ \mu(G) < \delta $ implies $ \int_G |f|^p d\mu < \epsilon $. By Egorov's theorem (applied using $ \delta $), there is a closed set $ F \subseteq [0,1] $ such that $ f_n \rightarrow f $ uniformly on $ F $ and $ \mu(X-F) < \delta $. We then have the estimates, $$ \int_X |f_n - f|^p d\mu = \int_F |f_n - f|^p d\mu + \int_{X-F} |f_n - f|^p d\mu \le \int_F |f_n - f|^p d\mu + \int_{X-F} (|f_n| + |f|)^p d\mu \le \int_F |f_n - f|^p d\mu + \left[\left(\int_{X-F} |f_n|^p d\mu \right)^{1/p} + \left(\int_{X-F} |f|^p d\mu \right)^{1/p} \right]^p $$ where the last inequality is by Minkowski. Now, by uniform convergence on $ F $, there exists $ N $ such that $ |f_n(x)-f(x)| < \epsilon $ for all $ n \ge N, x \in F $. Therefore, $ \int_F |f_n - f|^p d\mu \le \int_{F} \epsilon^p d\mu \le \epsilon^p $ for all $ n \ge N $. Since $ \mu(X-F) < \delta $, we have $ \int_{X-F} |f|^p d\mu < \epsilon $ and $ \left(\int_{X-F} |f_n|^p d\mu \right)^{1/p} \le ||f_n||_p \le M $ for all $ n \in \mathbb{N} $.
Combining, we get for all $ n \ge N $, $$ \int_X |f_n - f|^p d\mu \le \epsilon^p + (M + \epsilon^{1/p})^p $$ This estimate is not good enough; the $ M $ causes a problem. Is there a way around or another way?
Your strategy led you to try and prove $\|f_n-f\|_p \to 0.$ But this can fail. Example: Let $f_n(x) = n^{1/p}\cdot \chi_{(0,1/n)}.$ Then $f_n\to 0$ pointwise everywhere, and $\|f_n\|_p=1$ for all $n.$ Using Holder, check that $\int_0^1 f_n(x)g(x)\,dx\to 0$ for all $g\in L^q.$ But we clearly don't have $f_n\to 0$ in $L^p.$