## Appendix A Proofs of the Two Main Theorems
### Proof of the General Theorem
#### a.1.3 Verifying the Conditions of Proposition 1  
With Proposition 1, we prove the remaining points of Theorem 1: \((i)(b)\) and \((ii)\).  
For the application of Proposition 1:* \(Cl(\{\theta_{n}\}_{n})\) (set closure) plays the part of the compact \(K\)
* \(\{\theta\in\Theta|T(\theta)=\theta\}\) plays the part of the set \(\mathcal{L}\)
* \(\mathcal{L}\cap K\) is also compact thanks to hypothesis \(M3\)
* The likelihood \(g\) is the \(C^{0}\) Lyapunov function with regards to \((T,\mathcal{L})\)
* \(\{\theta_{n}\}_{n}\) is the \(K\) valued sequence (since \(K\) is \(Cl(\{\theta_{n}\}_{n})\)).  
The last condition that remains to be shown to apply Proposition 1 is that:  
\[\lim_{n\to\infty}\lvert g(\theta_{n+1})-g\circ T(\theta_{n})\rvert=0\,.\]  
We have more or less already proven that, in the previous section of the Proof, with \(F_{n}(\theta_{n})\) in place of \(\theta_{n+1}\). The only indices where \(F_{n}(\theta_{n})\
eq\theta_{n+1}\) are when the value of the sequence \(p_{n}\) experiences an increment of \(1\). We have proven with Proposition 2 that there is only a finite number of such increments.  
\[\lvert g(\theta_{n+1})-g\circ T(\theta_{n})\rvert=\lvert g(\theta_{0})-g\circ T (\theta_{n})\rvert\mathds{1}_{p_{n+1}=p_{n}+1}\,+\,\lvert g\circ F_{n}(\theta _{n})-g\circ T(\theta_{n})\rvert\mathds{1}_{p_{n+1}=p_{n}}\,.\]  
Since there is only a finite number of increments of the value of \(p_{n}\), then \(\exists N\in\mathbb{N}\), \(\forall n\geq N\), \(\mathds{1}_{p_{n+1}=p_{n}+1}=0\) and \(\mathds{1}_{p_{n+1}=p_{n}}=1\). In other words:  
\[\exists N\in\mathbb{N},\,\forall n\geq N,\,\,\,\lvert g(\theta_{n +1})-g\circ T(\theta_{n})\rvert=\lvert g\circ F_{n}(\theta_{n})-g\circ T( \theta_{n})\rvert\] \[\exists N\in\mathbb{N},\,\forall n\geq N,\,\,\,\lvert g(\theta_{n +1})-g\circ T(\theta_{n})\rvert=\lvert g\circ F_{n}(\theta_{n})-g\circ T( \theta_{n})\rvert\mathds{1}_{\theta_{n}\in Cl(\{\theta_{k}\}_{k})}\,.\]  
Since \(\theta_{n}\) is always in \(Cl(\{\theta_{k}\}_{k})\) by definition. Additionally Proposition 2 tells us that \(Cl(\{\theta_{k}\}_{k})\) is a compact. Moreover, in order to use Proposition 2 in the first place, we had proven that:  
\[\forall\text{ compact }K\subseteq\Theta,\quad\lim_{n\to\infty}\lvert g\circ F _{n}(\theta_{n})-g\circ T(\theta_{n})\rvert\mathds{1}_{\theta_{n}\in K}=0.\]  
We can apply this directly with \(K=Cl(\{\theta_{k}\}_{k})\) to conclude the desired result:  
\[\lim_{n\to\infty}\lvert g(\theta_{n+1})-g\circ T(\theta_{n})\rvert=0\]  
Hence we verify all the conditions to apply Proposition 1.  
Appendix A.1.4. Applying Proposition 1  
Since we verify all we need, we have the conclusions of Proposition 1:  
* \(\{g(\theta_{n})\}_{n\in\mathbb{N}}\) converges towards a connected component of \(g(\mathcal{L}\cap Cl(\{\theta_{n}\}_{n}))\subset g(\mathcal{L})\).
* If \(g(\mathcal{L}\cap Cl(\{\theta_{n}\}_{n}))\) has an empty interior, then \(\{g(\theta_{n})\}_{n\in\mathbb{N}}\) converges towards a \(g^{*}\in\mathbb{R}\) and \(\{\theta_{n}\}_{n}\) converges towards \(\mathcal{L}_{g^{*}}\cap Cl(\{\theta_{n}\}_{n})\). Where \(\mathcal{L}_{g^{*}}:=\{\theta\in\mathcal{L}|g(\theta)=g^{*}\}\).  
Both points are respectively the statements \((i)(b)\) and \((ii)\) of Theorem 1.  
Which concludes the proof of the Theorem.