Suppose $u$ vanishes on

$$\{x_{k+1}=...=x_{n}=0\}.$$

By subtracting a linear function of the form $a_{k+1}x_{k+1}+...+a_{n}x_{n}$ we may assume that $u(te_{n})=o(t)$. Then $S_{h,0}(0)$ has length $R(h)h$ in the $e_{n}$ direction, where $R(h)\rightarrow\infty$ as $h\rightarrow 0$. Furthermore, $S_{h,0}(0)$ has length exceeding $\frac{1}{C}h$ in the $e_{n-k},...,e_{n-1}$ directions, where $C$ is the Lipschitz constant of $u$ in $B_{1/2}$. Finally, $S_{h,0}(0)$ contains the unit ball in the subspace spanned by $\{e_{1},...,e_{k}\}$. We conclude that

$$|S_{h,0}(0)|\geq C^{-k}R(h)h^{n-k},$$

which contradicts Lemma 2.2 as $h\rightarrow 0$ for $k\geq\frac{n}{2}$. ∎