Which ones will be correct and why?
If continuous functions ${f_n}$ converge to continuous function $f$ point wise on $\mathbb{R}$ .
$1)$$0 \leq f_n\leq f$ then $\lim_{n\rightarrow \infty}\int_{-\infty}^{\infty} f_n(x) = \int_{-\infty}^{\infty}f(x)$
$2)$ $|f_n(x)|\leq |\sin(x)|$ for all $x \in \mathbb{R}$ then $\lim_{n\rightarrow \infty}\int_{-\infty}^{\infty} f_n(x) = \int_{-\infty}^{\infty}f(x)$
$3)$ $|f_n(x)|\leq e^x $ for all $x \in \mathbb{R}$ and $a$ less than $b$ and $a,b \in \mathbb{R}$ then $\lim_{n\rightarrow \infty}\int_{a}^{b} f_n(x) = \int_{ a}^{b}f(x)$
All I know Uniform convergence alone can not permit the limit to get into the integration. But How I would proceed for this case I have no idea. Can anyone help me out?