Set $f_n= n1_{[0,1/n]}$
For $0 , 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.