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 $L^p$?
As discussed in the comments:
Remember that compactly supported functions are dense in $L^p$. Then $C(G)$ is dense in both $L^2$ and $L^p$. So pick any $f$ in $L^2$ and a sequence $g_n$ of continuous functions (yep, since $G$ is compact they're automatically compactly supported) converging to $f$ in $\|\cdot\|_{L^2}$. Then $g_n$ is in $L^p$ but its limit is not.