\begin{align*}
u(s,h_{1/\sqrt{\lambda}}) + \lambda u(S,h_{-\sqrt{\lambda}}) &= 1+\lambda + \varepsilon n (1+\lambda) + 2\varepsilon \cdot \sum_{i \text{ even, } 0 \le i \le n-2} \frac{\lambda x_i}{x_{i-1}} +  2\varepsilon \cdot \sum_{i \text{ odd, } 1 \le i \le n-1} \frac{x_i}{x_{i-1}} \\
& \ge 1+\lambda + \varepsilon n(1+\lambda) + 2 \varepsilon n \left( \lambda^{n/2} \prod_{i=0}^{n-1} \frac{x_i}{x_{i-1}} \right)^{1/n},
\end{align*}
