Can you please help me solve this on measure theory? My TA did not go over this. He said we are not going over this but you can do this if you want. Can someone please explain to me? Thanks.
Suppose $f_n\to f$ on $[1,\infty)$ and $|f_n(x)| \le 1 / x^2$ where $1\le x$.
Is it true that $ \int_{[1,\infty)} f_n\to \int_{[1,\infty)} f$ ?
you have $|f_n (x)|\leq \frac{1}{x^2}$ , for $x \geq 1$.
By the dominated convergence theorem you have
$$\lim_{n \rightarrow \infty } \int f_n = \int \lim_{n \rightarrow \infty } f_n = \int f .$$ In the first equality was use the dominated convergence theorem