I am working on a problem$^{(1)}$ similar to this 2013 posting:
Suppose that $f_n$ is a sequence of integrable, non-negative functions, so that $\forall x$, $f_n(x)$ decreases to $f(x)$. Show the following is true: $$\int f_n d \mu \rightarrow \int f d \mu.$$
Aside from the apparent differences between the two questions, this problem comes from early chapter of Measure & Integration class, so that Monotone Convergence Theorem is not in the background. Here are what I have been attempting to do $-$ rightly or wrongly:
Since $f_n(x)$ decreases to $f(x)$, this implies that $$\lim_{n \to \infty} f_n(x) = f(x). \tag{1}$$
Taking integral on both sides, $$\begin{align} \int \lim_{n \to \infty} f_n(x) d \mu = \int f(x) d \mu \tag{2}\\ \lim_{n \to \infty} \int f_n(x) d \mu = \int f(x) d \mu \tag{3}\\ \end{align}$$
implying that
$$\int f_n d \mu \rightarrow \int f d \mu. \tag{4}$$
But I am not sure if the move from (2) to (3) is valid. Please let me know what I should do instead. Thank you for your time and help.
(1) Richard F. Bass' Real Analysis, 2nd. edition, chapter 6: The Lebesgue Integration, Exercise 6.6, page 50.