This ensures that for all hyperedges $e_j \in E$ and vertices $i \in e_j$, the difference $x_i - z_j$ remains the same, which in turn leaves the first term of unchanged. For the second term of , we find that the difference $x_i - x_j$ only changes if $i \in V_1$ and $j \in V_2$, or vice versa.

For all $1 \leq i \leq k'$, $k' < j \leq k$, we have $$\lim_{r \rightarrow \infty} \frac{w_i \, w_j}{\|2 \, (r \, y) + x_i - x_j\|^{\delta}} 
= \lim_{r \rightarrow \infty} \frac{1}{|r|^{\delta}} \, \frac{w_i \, w_j}{\|2 \, y + (x_i - x_j)/r\|^{\delta}}
= 0$$ as $y \neq 0$ and $\delta > 0$. In particular there exists an $r > 0$ such that for all $1 \leq i \leq k'$, $k' < j \leq k$ we have $$\smash{\frac{w_i \, w_j}{\|2 \, (r \, y) + x_i - x_j\|^{\delta}} - \frac{w_i \, w_j}{\|x_i - x_j\|^{\delta}} < 0.}$$ So for this $r$ we have $$f(x_1 + r \, y, \ldots, x_{k'} + r \, y, x_{k' + 1} - r \, y, \ldots, x_k - r \, y) < f(x_1, \ldots, x_k),$$ hence $(x_1, \ldots, x_k)$ does not have minimum energy; we have reached 