Existence of a 0- and 1-valent configurations in the proof of FLP impossibility result

Question

In the known paper Impossibility of Distributed Consensus with one Faulty Process (JACM85), FLP (Fisher, Lynch and Paterson) proved the surprising result that no completely asynchronous consensus protocol can tolerate even a single unannounced process death.

In Lemma 3, after showing that D contains both 0-valent and 1-valent configurations, it says:

Call two configurations neighbors if one results from the other in a single step. By an easy induction, there exist neighbors C₀, C₁ ∈ C such that Dᵢ = e(Cᵢ) is i-valent, i = 0, 1.

I can follow the whole proof except when they claim the existence of such C₀ and C₁. Could you please give me some hints?

Answer 1

D (the set of possible configurations after applying e to elements of C) contains both 0-valent and 1-valent configurations (and is assumed to contain no bivalent configurations).

That is — e maps every element in C to either a 0-valent or a 1-valent configuration. By definition of C, there must be a root element that is connected to all other elements by a series of "neighbour" relationships, so there must be a boundary point where an element in C that leads to a 0-valent configuration after e is neighbours with an element in C that leads to a 1-valent configuration after e.