I'm conjecturing that with Complete-linkage clustering two elements from the same cluster will always be closer to each other some other element from another cluster.
In more formal terms:
Let $C$ be a clustering. $\not\exists z \in C_j$ s.t. $\bigtriangleup(x, z) < \bigtriangleup(x, y)$ where $x,y \in C_i$, $C_i \neq C_j$ and $C_i, C_j \in C$.
I haven't been able to prove the conjecture yet, thus I'm wondering whether I'm right or wrong. If this is indeed the case, I would much appreciate a sketch a proof. I'm pretty sure I can work my way from there.
On a side note (not that I think it makes a difference), I'll be applying the clustering algorithm on a one-dimensinal dataset.
Your input is much appreciated.