Counterexample convergence of a sequence of functions in the $L^1$ norm vs pointwise I have to show that $f(x) \neq \lim_{n\to\infty} f_n(x_n) $, where $x_n$ is a converging sequence in $[0,1]$, and $f_n \rightarrow f$ in the $L^1$ norm, such that $ \int_0^1 | f_n (x) - f(x) | dt < \epsilon $.
So, as far as I got, I need to find a function which does not converge pointwise, while it converges in the norm.
Getting examples from uniform vs pointwise convergence is quite easy, but I cannot work this out: it's like if I have to come out with a function whose integral e.g. goes to zero but the function moves around in the interval.
I'm looking for a countrexample to something like stated here.

Take for example $f_n(x)=x^n$. Then $f_n\to f=0$ in $L^1([0,1])$: 
$$\int_0^1|f_n(x)-f(x)|dx=\int_0^1 x^n dx=\frac{1}{n+1}\to 0.$$
On the other hand, $x_n=1-\frac{1}{n^2}\to 1$ and
$$\lim_{n\to\infty}f_n(x_n)=\lim_{n\to\infty}\left(1-\frac{1}{n^2}\right)^n=1\not=f(1)=0.$$
