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Then $\O$ does not converge in UP.  \end{lemma}  \begin{proof}  Let $G$ be a (serial and non-necessarily functional) influence profile, and $\O$ be an opinion profile, such that, for some $p\in \Atoms $, for all $i,j\in C$ with $C$ a connected component of $G_p$: if $i\in R_p(j)$, then $O_i(p)\neq O_j(p)$. Then, by definition of UPs, for all $i\in C$, $O^1_i(p)\neq O_i(p)$, and by repeting repeating  the same argument, for all $n\in\mathbb{N}$, $O^{n+1}_i(p)\neq O^n_i(p)$. \end{proof}  Recall that a graph is properly colored if each node is assigned exactly one color and no node has a successor of the same color, and consider the two possible opinions on issue $p$ as colors. The above result can be reformulated as follows: if for some $p\in\Atoms$, $\O$ properly colors $G_p$, then $\O$ does not converge. In fact, it is easy to see that in such a case, all agents will change their opinion on $p$ at every step, entering an oscillation of size $2$. Note that this limit case of opinion distribution is yet another special case of DeGroot processes, another example within the intersection between the two frameworks of propositional opinion diffusion and DeGroot.