dominated convergence theorem

i dont see why $\displaystyle g_n$ has to converge uniformly to $\displaystyle g$. to me if $\displaystyle g_n \rightarrow g$, then $\displaystyle g_nf \rightarrow gf$(i am not sure how to show). and $\displaystyle |g_nf| \leq Mf$ since there exists $\displaystyle M $ such that $\displaystyle |g_n| \leq M$ and since $\displaystyle f$ is integrable, so is$\displaystyle Mf$.

by dominated convergence theorem, i think we get what we want.

can someone help me with this one?