Under the assumptions of Proposition 6.40, prove the following claims:

(i) a rational bubble cannot restart once it bursts, i.e., \(\beta_{t}=0\) implies that \(\beta_{t+s}=0\) for all \(s \in \mathbb{N}\);

(ii) suppose that a price process is uniformly bounded from above, i.e., there exists a constant \(K>0\) such that \(\mathbb{P}\left(s_{t} \in[0, K]\right)=1\) holds for all \(t \in \mathbb{N}\). Prove that \(s_{t}^{*}=s_{t}\) holds for all \(t \in \mathbb{N}\), i.e., there is no bubble component.