I was comparing my solution (bottom) to the following question with the solution posted here, and I fail to understand why we require Fatou's lemma at all, and not just linearity and monotonicity of integration?
Let $f_n:R\rightarrow R$ be a sequence of nonnegative measurable functions that converges pointwise to an integrable function $f$. Show that if $$\int\limits_{R}f=\lim\limits_{n\to\infty}\int\limits_R f_n$$ then $\int\limits_{E}f=\lim\limits_{n\to\infty}\int\limits_E f_n$ For any measurable set $E$.
Hint: Use the Fatou's Lemma twice.
Why is it not just using the formal limit definition as such:
For every $E$ and $k\in \mathbb{N}$, we have:
$|\int_E f - \int_E f_k| = |\int_E (f-f_k)| \leq |\int (f-f_k)| = |\int f - \int f_k|$
Now, since $\lim ∫f_k=∫f$, for every $\epsilon>0$ there is a $k$ large enough so that RHS is $\leq \epsilon$, and therefore so is $|\int_E f - \int_E f_k|<\epsilon$.
What am I missing? It clearly states "Show that if...", so are we not supposed to assume that what is immediately after the "if" is a given?
Thanks for any help.
