Euality between these two L1 integrable functions?

Question

Let $$\lim_{n\to\infty}\int |u_n v_n-uv| d \mu=0$$

I want to show that

$$\lim_{n\to\infty}\int u_n v_n d \mu=\int u v d \mu$$

What I have so far: $$\lim_{n\to\infty}\int u_n v_n d \mu-\int u v d \mu$$$$=\lim_{n\to\infty}\int u_n v_n d \mu-\lim_{n\to\infty}\int u v d \mu $$$$= \lim_{n\to\infty}\int u_n v_n-uv d \mu$$$$\le \lim_{n\to\infty}\int |u_n v_n-uv| d \mu=0$$

Now I'm not sure how to show that the difference is non-negative.

Answer 1

$$\left|\int u_n v_n d \mu -\int uv d \mu\right| \leq\int |u_n v_n-uv| d \mu$$