In Chapter 5 (Sobolev Spaces) of Partial Differential Equations (2nd Edition) by Evans, Example 4 says the following:
Let $\{r_k\}_{k=1}^\infty$ be a countable, dense subset of $U=B^0(0,1)$. Write $$ u(x)=\sum_{k=1}^\infty \frac{1}{2^k}\vert x-r_k \vert^{-\alpha}, \quad x\in U $$ Then $u\in W^{1,p}(U)$ for $\alpha <\frac{n-p}{p}$.
I am having trouble proving that $u\in W^{1,p}(U)$. I have tried looking at the partial sum $$u_n(x)=\sum_{k=1}^n \frac{1}{2^k}\vert x-r_k \vert^{-\alpha}$$ and attempt to show that $(u_n)$ is a Cauchy sequence in $W^{1,p}(U)$. However, I am not able to make any progress in showing that $(u_n)$ is indeed a Cauchy sequence. Any advice on how I can show that this gives a Cauchy sequence?