$\{f_n\}$ integrable functions and $\{g_n\}$ uniformly bounded sequence of continuous functions, then is the following statement true?

Question

Let $\{f_n\}_{n=1}^\infty$ and $f$ be integrable functions on [0,1] such that

$\displaystyle\lim_{n \rightarrow \infty} \int_{0}^1 |f_n(x) - f(x)| dx = 0$.

Then, is the following statement true?

If $\{g_n\}$ is uniformly bounded sequence of continuous functions converging pointwise to a function $g$ , then,

$\displaystyle\int_0^1 |f_n(x) g_n(x) - f(x) g(x) | dx \rightarrow 0 $ as $ n \rightarrow \infty$.

Could you please help me solve this problem?

Answer 1

Notice that \begin{align*} \int | f_ng_n -fg| &\leq \int | f_n g_n - fg_n| + \int |f g_n -fg|\\ & \leq M \int |f_n-f| + \int |f| |g_n-g| \end{align*} where $M = \sup ||g_n||_\infty$. Furthermore, $| g_n -g| |f| \to 0$ almost everywhere, and it is dominated by an integrable function. Can you finish the proof? (apply dominated convergence thm)