= Solution
The <dominated convergence theorem> states: if measurable $f_n$ converge almost everywhere to $f$ and $|f_n|\le g$ almost everywhere for an integrable $g\ge0$, then $f$ is integrable and $\int f_n\to\int f$. In fact $\int|f_n-f|\to0$.
To derive it from the <monotone convergence theorem>, first prove <Fatou's lemma>. For nonnegative measurable $h_n$, the functions $k_N=\inf_{n\ge N}h_n$ increase to $\liminf h_n$ and satisfy $\int k_N\le\inf_{n\ge N}\int h_n$. <Monotone convergence> yields $\int\liminf h_n\le\liminf\int h_n$.
Now $|f|\le g$ almost everywhere, so $f$ is integrable. Apply <Fatou's lemma> to the nonnegative functions $2g-|f_n-f|$, which converge almost everywhere to $2g$:
$$
2\int g\le2\int g-\limsup_n\int|f_n-f|.
$$
Hence $\int|f_n-f|\to0$, and $|\int f_n-\int f|\le\int|f_n-f|$ proves the assertion. Common exceptional null sets may be removed at the beginning since the sequence is countable.
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