Cauchy but not convergent sequence in $L^1$

Question

Show that the $\arctan(nx)$ is Cauchy but not convergent in the $1$-norm of the space $C^1 [-1,1]$, where the standard $1$-norm is $||f||_1 = \int \limits _0^1 |f(x)| dx$

Answer 1

Hint: Write $f_n(x) = \arctan (nx)$. Try to show that $\{f_n\}$ converge in $L^1$ norm to the function

$$f(x) = \begin{cases} \pi/2 &\text{if } x\in [0,1], \\-\pi/2 &\text{if } x\in [-1,0). \end{cases}$$

That is, try to show

$$ \int_{-1}^1 |f_n(x) - f(x)| dx \to 0$$

as $n\to \infty$. You might consider splitting the integral into $[-1,0]$ and $[0,1]$ part.