I'm working on the following problem:
Given a sequence of integrable functions $f_n: \mathbb{R} \to \mathbb{R}$ with $f_n \to 0$ pointwise and $|f_n(x)|≤ \frac{1}{|x|+1}$ for all $x$ and $n≥1$, prove or find a counterexample of the following assertion: $$\lim_{n \to \infty}\int_{-\infty}^{\infty} |f_n(x)| dx=0$$
I'm thinking that this is false as the function $\frac{1}{|x|+1}$ may not dominate $|f_n(x)|$ for all $x$, but I'm not sure what sequence of $f_n$'s would serve as a counterexample here.
As $$\int_0^\infty\frac{dx}{|x|+1}=\int_0^\infty\frac{dx}{x+1}=\infty,$$ for all $n$, there is an $a_n>n$ with $$\int_n^{a_n}\frac{dx}{x+1}=1.$$ Let $f_n(x)$ equal $1/(x+1)$ on the interval $[n,a_n]$ and $0$ elsewhere.