Proof of Theorem 1

[fkcptlst]

Hello authors, thanks for your insightful work.

I have some questions regarding proof of theorem 1:

Eq. 16 says $\Delta x_t$ is negligible. Why exactly is this the case? Since $\Delta \epsilon_t$ is not negligible and $\frac{\sqrt{1 - \alpha_t - \beta_t} - \sqrt{1 - \alpha_t}}{\sqrt{1 - \beta_t}}$ is not necessarily zero.