Sequences and integration problem Is it true that 2 sequnces of real numbers $a_n$ and $b_n$ converge to 0 if   $\lim\limits_{n\to\infty}\int_{0}^{1}|a_nx+b_n|dx=0$ ?
I tried to relate to the integral to see if I get to some result, but it didn't worked out. I learned integrals and also limits and the problem it is from a book I'm working from.
 A: Yes! Notice that $|a_n + b_n|$ needs to converge to $0$. If it doesnt, then ther would need to be some $\varepsilon > 0$ such that $|a_n + b_n| > \varepsilon$ for some $n_0\in\mathbb{N}$ and all $n>n_0$. Since the function $|a_nx + b_n|$ is continuous  there exists some $\delta > 0$ such that $|a_nx + b_n| > \varepsilon/2$ for all $x\in(1-\delta,1]$ This means that
$$
\int_0^1 |a_nx + b_n|dx \geq \int_{1-\delta}^1 |a_nx + b_n|dx \geq\frac{\varepsilon\delta}{2} > 0
$$
Which would imply that the sequence of integrals cant converge to $0$. This means that
$$
\lim_{n\to\infty}(a_n+b_n) = \lim_{n\to\infty}|a_n+b_n|=0
$$
Using a similar approach you can show that the sequence $|\frac{a_n}{2}+b_n|$ also needs to converge to zero which implies that $\lim_{n\to\infty}(a_n/2+b_n)=0$. Multiplying both sides by $2$ you get.
$$
\lim_{n\to\infty}(a_n+2b_n)=0
$$
Subtracting the bottom equality from the top one you get
$$
\lim_{n\to\infty}b_n = \lim_{n\to\infty}(a_n + 2b_n) - \lim_{n\to\infty}(a_n + b_n) = 0
$$
from which you can also deduce that $a_n$ has to go to $0$ aswell.
