= Solution
Now $f\ge0$ <almost everywhere>. The functions $\min(f_n,f)$ converge <almost everywhere> to $f$ and are dominated by the integrable $f$. The <dominated convergence theorem> gives $\int\min(f_n,f)\to\int f$. Using $|f_n-f|=f_n+f-2\min(f_n,f)$ and the assumed convergence of the nonnegative integrals yields
$$
\boxed{\int|f_n-f|\,d\mu\longrightarrow0.}
$$
This is the nonnegative form of <Scheffé's lemma>: almost-everywhere convergence together with convergence of total mass implies convergence in the <L1 norm>.
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