From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ be a finite measure space that is also a compact topological group and let $p>2$.
Now endow $L^p(G)$ with the $L^2$ norm.
Could someone give me an example of a Cauchy sequence in $L^p$ such that its limit s not in $L^p$?