Proof of Vitali's Convergence Theorem

Solution 1:

Your first two steps are fine to me (Use of uniform integrability and Egoroff's Theorem).

Note that in general if $f_n\to f$ and $\int_E |f_n|\leq M$ for some $M$, by Fatou's Lemma you have $$ \int_E |f|= \int_E \liminf |f_n| \leq \liminf \int_E |f_n|\leq M. $$

To finish your proof you must say:

So, for any $n\geq N$ (the $N$ in your post) $$\begin{align*} \int_X |f_n-f|~d\mu & = \int_{E} |f_n-f|~d\mu +\int_{E^c} |f_n-f|~d\mu\\ & \leq \int_{E} |f_n-f|~d\mu + \int_{E^c} |f|~d\mu + \int_{E^c} |f_n|~d\mu\\ & \lt \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3}\\ & =\varepsilon. \end{align*}$$ The second step is justified by the triangle inequality and the observation made at the beginning of this post.

To justify that $f\in L^1$, see Nate's comment.