I am currently self studying Mathematical analysis by M. Apostol. I got stuck in trying to understand
$\\$ Theorem 10.30 $\ \ $Let ${f_n}$ be a sequence of functions in $L(I)$ which converges almost everywhere on $I$ to a limit function $f$. Assume that there is a nonnegative function g in $L(I)$ such that $$|f(x)| < g(x) \text{ a.e. on I}$$ Then $f\in L(I)$.
$\\$ Proof $\\$ Define a new sequence of functions $g_n$ on $I$ as follows : $$g_n = \max \{\min (f_n, g), -g\}$$ $\dots$ Then $|g_n (x)|\leq g(x)$ almost everywhere on $I$, and it is easy to verify that $g_n \to f$ almost everywhere on $I$. Therefore, by the Lebesgue dominated convergence theorem, $f \in L(I)$.
Can someone please highlight how $g_n \to f$?
Let $\Omega $ a set of measure $0$ s.t. $|f(x)|<g(x)$ for all $x\notin \Omega $. Let $x\notin \Omega $. Since $f_n(x)\to f(x)$, there is $N\in \mathbb N$ s.t. for all $n\geq N$, $$|f_n(x)|<g(x).$$ In particular, if $n\geq N$, $$g_n(x)=\max\{f_n(x),-g(x)\}=f_n(x).$$