Understanding convergence in $L^P$ ( Through a counter example )

Question

Let $1<p_0<\infty$. I am looking to find an example of a sequence $\{f_k\}_{k\in \mathbb N}$ such that ${f_k}\in L^P$ for $1 \leq p < \infty$ , $ f_k\xrightarrow {}0$ in $L^P$ , for $1<p<p_0$ , but $f_k$ does not converge in $L^{p_0}$