§5.2, p. 1585 — two-sided bounds on the destabilizing delays τ_{n,l} (corrected)

Let $0 < \mu_1 < \mu_2$, $l \in \mathbb N$, $\Lambda > 0$, $0 < \alpha < \tfrac12(\mu_2 - \mu_1)$ and $\tau > 0$ with

Then

1. $\displaystyle \tau > \frac{(2l+1)\pi}{\Lambda}$;
2. if moreover $\Lambda^2 > \tfrac14(\mu_2 - \mu_1)^2$, then

Applied to the delays $\tau_{n,l}$ of Case (b), bound 1 shows $\tau_{n,l} \to +\infty$ as $l \to \infty$ for a fixed eigenvalue $\Lambda_n^2$, and bound 2 shows $\tau_{n,l} \to 0^+$ as $\Lambda_n \to \infty$ for a fixed $l$: the destabilizing delays are arbitrarily large and arbitrarily small.

Formalization Note The paper derives both limits from the single inequality $(2l+1)^2\pi^2/\tau_{n,l}^2 \le \Lambda_n^2$, which bounds $\tau_{n,l}$ only from below and so gives only the limit $l \to \infty$. The upper bound 2, which uses $\alpha < \tfrac12(\mu_2 - \mu_1)$, is the step needed for $\tau_{n,l} \to 0^+$; the statement includes it as the corrected form of the page's argument.