 Solution

Because $U\in [0,1]$, the conditional acceptance probability is $a_k=\min (1,{\lambda }^k)$, even when the printed threshold exceeds one. Repeating rejected trials gives the clipped acceptance reweighting identity:For $0<\lambda <1$, $a_k={\lambda }^k$, including $a_0=1$. The denominator is $e^{-\lambda }{\sum }_k{\lambda }^{2k}/k!=e^{{\lambda }^2-\lambda }$, soFor $\lambda \ge 1$ every proposal is accepted, so reweighting by ${\lambda }^k$ would be incorrect. At $\lambda =1$ both formulas agree. Termination has probability one since the acceptance probability is $e^{{\lambda }^2-\lambda }>0$ in the first case and one in the second.