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.