Show a sequence of densities converge to some density.

Question

Consider a sequence of density functions $f_{n}(x)$ and another density function $g(x)$, I was wondering to show $$\lim_{n\to\infty}\int |f_{n}(x)-g(x)|dx=0,$$ is it enough to show that $\lim_{n\to\infty}f_{n}(x)=g(x)$? It seems that $$|f_{n}(x)-g(x)|<f_{n}(x)+g(x)$$ and for any $n$ $$\int f_{n}(x)+g(x)dx=2,$$ so dominated convergence theorem applies?