Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be integrable function in every closed and bounded interval.
Assuming $ \int _{0}^{\infty}|f(t)|\,dt \ \ $ exists.
I have to prove that there exists a sequences of $x_n \in \mathbb{R}$ so that
$$\lim_{n\to\infty}x_n=\infty \quad\text{and}\quad\lim_{n\to\infty}f(x_n)=0$$
So how do I do that ?
What happen if I only know that $ \int _{0}^{\infty}f(t)\,dt \ \ $is exists and $f$ is continues function
any help will be appreciated
Thanks in advanced !!
Hint: Note that $|f|$ is also integrable. Let $I_n=\int_0^n|f(x)|\,\mathrm dx$. By assumption, $(I_n)_{n\in\mathbb N}$ is a Cauchy sequence. Therefore $\int_n^{n+1}|f(x)|\,\mathrm dx\to 0$. Conclude that you can pick a suitable $x_n\in [n,n+1]$.