Given $(\mathbb{R},S,\mu)$, where $S$ is the $\sigma$-field of Borel subsets of $\mathbb{R}$, $\mu$ the Lebesgue measure. On the vector space of integrable simple functions (two simple functions are equivalent if they differ on at most a null set), give the norm $\lVert f \rVert_1=\int |f(x)|d\mu (x)$. I want to construct a sequence of functions that is Cauchy for this norm but does not converge for the norm.
Thanks in advance.