Convergence in $L^{p}$ spaces Set $$f_n= n1_{[0,1/n]}$$
For $0<p\le\infty $ , one has that  $\{f_n\}_n$ is in $L^p(\mathbb R)$. For which values of $p$ is  $\{f_n\}_n$ a Cauchy sequence in $L^p$? Justify your answer.
This was a Comp question I was not able to answer. I don't mind getting every details of the proof.
What I know for sure is for $p=1$,  $\{f_n\}_n$ is Cauchy in $L^p$ because when you get the integral of the function that is going to equal 1 no matter the value of $n$. So the sequence is not convergent in $L^1$, and hence is not Cauchy. I do not know how can I be more rigorous on this problem.  Any help much appreciated. 

You need to compute $\int_0^1 |f_n(x) - f_m(x)|^p dx$ by hand. For $p \ge 1$, this is not a Cauchy sequence, since $\Vert f_n \Vert_{L^p}^p = n^{p-1}$. For $p < 1$ there is hope :)
