$x^i_0 = p$ and $x^i_1 = q$ from some step, whence $\{p,q\} \in {\mathscr{X}}$ and condition PC1 is clearly satisfied with $x'^j_0 = p$ and $x'^j_1 = q$ for each $j \in {\mathbb{Z}}_{\geqslant 0}$.

We can thus assume that $p \in L({\mathscr{Z}})$ or $q \in L({\mathscr{Z}})$.

Then by passing to a subsequence we may assume $$x^i_0 \leqslant p \leqslant {x^i_1}^{--} < x^i_1 \leqslant q \leqslant {x^i_0}^{--}$$ for each $i \in {\mathbb{Z}}_{\geqslant 0}$. Let $Y = \{y_0,y_1\}$ be a diagonal of ${\mathscr{Z}}$ with $$p \leqslant y_0 \leqslant {x^i_1}^{--} \text{ and } q \leqslant y_1 \leqslant {x^i_0}^{--}$$ for each $i \in {\mathbb{Z}}_{\geqslant 0}$, see Figure \[fig:ys\]. Note that such diagonals exist; in fact since ${\mathscr{Z}}$ satisfies the two-sided limit condition (see Definition \[D:admissible\]), we can even find an entire sequence of such diagonals with endpoints converging to $p$ and $q$ (at least one of which lies in $L({\mathscr{Z}})$) from above.

(0,0) circle(0.5cm); (60:0.5) node\[fill=white,circle,inner sep=0.065cm\] circle (0.02cm); (220:0.5) node\[fill=white,circle,inner sep=0.065cm\] circle (0.02cm); at (60:0.6)[$q$]{}; at (220:0.6)[$p$]{};

at (85:0.6)[$y_1$]{}; (85:0.48) – (85:0.52); at (295:0.6)[$y_0$]{}; (295:0.48) – (295:0.52); (85:0.5) .. controls (85:0.2) and (295:0.2) .. (295:0.5); at (0:0.6) [$x^i_1$]{}; (0:0.48) – (0:0.52); at (160:0.6) [$x^i_0$]{}; (160:0.48) – (160:0.52); (0:0.5) .. controls (0:0.2) and (160:0.2) 