Doubt in Scheffe's Lemma

While reading up on "Glivenko Cantelli Theorem" from Probability Models by K.B Athreya, the author used 2 lemmae to prove it. One was called Scheffe's lemma, the other Polya's theorem.

Scheffe's Lemma is stated as follows:

Let $f_n, f$ be non negative $\mu$ integrable functions. If $f_n \to f$ a.e and $\int f_n d\mu \to \int fd\mu$, then $$\int |f_n - f|d\mu \to 0$$

My proof is:

Let $g_n = |f_n -f|$. Now we have $g_n \to 0$ a.e. Now $$0 \leq g_n = |f_n -f| \leq f + f_n$$ $$\Rightarrow \int g_n d\mu \leq \int fd\mu + \int f_nd\mu < \infty $$

Thus by Dominated convergence theorem, $$\int g_nd\mu \to 0$$ QED.

The Question: My doubt is that in this proof, I have not used that $\int f_n d\mu \to \int fd\mu$, at least not explicitly. So is this condition superfluous?

What I searched: I searched for Scheffe on MSE, but got 3 results (none useful) and when I typed Scheffe's instead, I got a result not belonging to these 3 which was actually on Scheffe's lemma (Though not helpful). It's strange (not the result but the search).

I'd appreciate any help/hints on this. Kindly ask me for clarifications if required.

Let

$$u_n = \max\{ f, f_n\} \quad \text{and} \quad l_n = \min \{ f, f_n \} $$

so that both $(u_n)$ and $(l_n)$ converge pointwise to $f$, $l_n \leq f \leq u_n$ and $|f - f_n| = u_n - l_n$. By DCT, it is clear that $\int l_n \, d\mu \to \int f \, d\mu$. thus from

$$ \int u_n \, d\mu = \int (f + f_n - l_n) \, d\mu ,$$

taking $n\to\infty$, we have $\int u_n \, d\mu \to \int f \, d\mu$. (Here the assumption that you are concerning is used.) Therefore we have

$$ \int |f - f_n| \, d\mu = \int u_n \, d\mu - \int l_n \, d\mu \to 0. $$

Observe $|f| + |f_n| - |f-f_n|\geq 0$, with Fatou's lemma $$\liminf \int |f| + |f_n| - |f-f_n| \geq 2 \int |f| $$ by the assumption $\lim \int |f_n| = \int |f|$, the left-hand-side of above inequality also equals $$\liminf \int |f| + |f_n| - |f-f_n| = 2\int |f| -\limsup \int |f-f_n|,$$ combine them we get $$2\int |f| -\limsup \int |f-f_n|\geq 2 \int |f| $$ $$\limsup \int |f-f_n| \leq 0 .$$

I don't think your "proof" works. Let $f_{n}=n\chi[0,\frac{1}{n}]$, $f=0$. Then $f_{n}\rightarrow f$ almost everywhere. But you cannot conclude that $$\int f_{n}d\mu\rightarrow 0$$ as it is constant $1$. As the commenter pointed out to apply dominated convergence theorem you need a function $g$ such that $|f_{n}|\le |g|$ and $g\in L^{1}(\Omega)$.