Not sure how to start proving that $\int_E f\,d\lambda$ = $\lim_{n \to \infty} \int_{E_n} f\,d\lambda$ Here is the following problem:

Suppose that $f$ is Lebesgue integrable over E and that $\{E_{n}\}_{n=1}^{\infty}$ is a increasing sequence of lebesgue measurable sets, that is, $E_1$ $\subset$ $E_2$ $\subset \cdots$ and $\cup_{n=1}^{\infty} E_{n}$ := E.  Prove that
$$\int_E f\,d\lambda = \lim_{n \to \infty} \int_{E_n} f\,d\lambda$$

Any suggestions/ hints on how to start this problem would be appreciated.
Hint: Let $f_n := f\chi_{E_n}$, where $\chi$ is the characteristic function. Then $f \chi_E = \lim_{n\to\infty} f_n$. Hence you would like to move the limit inside the integral, which theorem would be useful for that?
Case 1: Let $ f \ge 0$.

$$\lim_{n \to \infty} \int_{E_n} f\,d\lambda$$

$$\lim_{n \to \infty} \int_{E_n} f\,d\lambda$$

$$ = \lim_{n \to \infty} \int_{\Omega} f1_{E_n}\,d\lambda$$

$$ = \int_{\Omega} \lim_{n \to \infty}  f1_{E_n}\,d\lambda \tag{1}$$
$$ = \int_{\Omega} f 1_{E}\,d\lambda \tag{2}$$
$$ = \int_{E} f \,d\lambda$$

$(1)$ Now, try out: DCT? MCT? ICT? VCT? Scheffe's Lemma?

$(2)$ Good ol' $\varepsilon-N$ definition.



Case 2: Now, write $f = f^+-f^-$

Replace all $f$'s in Case 1 with $f^+-f^-$.