An example of a particular sequence of Riemann integrable functions

Question

I'm trying to understand better the different ways one can have a sequence of Riemann integrable functions.

Is it possible to have a sequence $(f_n)_{n=1}^{\infty}$ of Riemann integrable functions on $[0,1]$ such that:

$$\lim_{n\to\infty}\int_{0}^{1}f_n(x)dx = 0 \enspace\text{and}\space \lim_{n\to\infty}f_n(x)\space \text{does not exist for any}\space x \in [0,1]?$$

An example would be a great help.

Answer 1

Hint: take a sequence of functions of the form $$f_n(x) = \cases{1 & if $a_n \le x \le b_n$\cr 0 & otherwise\cr}$$ for suitable choice of $a_n$ and $b_n$.

Answer 2

Let $f_n(x)=\sin(nx)$, then $\lim\limits_{n\rightarrow +\infty}f_n(x)$ doesn't exist (except for $x=0$), but $$ \int_0^1f_n(x)dx=\frac{1-\cos(n)}{n}\underset{n\rightarrow +\infty}{\longrightarrow}0 $$