## $\{u_m\}$ is bounded in $W^{1, p}(U)$, but does not possess a (norm-)convergent sequence in $L^{p^∗}(U)$

Let $$U=B_1(0)$$ be the unit ball in $$\Bbb{R}^n$$, $$1≤p, $$p^∗=\frac{np}{n−p}$$. Consider the sequence:

$$$$\label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} u_m = \left\{ \begin{array}{ll} m^{\frac{n}{p}-1}(1-m|x|) & \mathrm{if\ } x <1/m \ 0 &\mathrm{if\ } x\geq 1/m\ \end{array} \right.$$$$ Prove that $$\{u_m\}$$ is bounded in $$W^{1, p}(U)$$, but does not possess a (norm-)convergent sequence in $$L^{p^∗}(U)$$.

I’ve managed to prove that $$\{u_m\}$$ is bounded in $$W^{1, p}(U)$$, and I’ve tried to prove the non-existance of a convegent subsequence via completeness of $$L^{p^∗}(U)$$ and taking a Cauchy series. So far I have gotten nothing. Any hints?