\in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{Z}}$ has a convergent subsequence $\{z'_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ converging to some point in $\overline{{\mathscr{Z}}}$.

There is hence a dichotomy:

-   Either the subsequence $\{z'_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ converges to $z \in {\mathscr{Z}}$, and $z'_i$ is constant from some step,

-   or the subsequence $\{z'_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ converges to a proper limit point $a \in L({\mathscr{Z}})$ and $z'_i$ is not constant from any step.

In the latter case, by refining the sequence further if necessary, we can suppose that the sequence is increasing (i.e. $z'_0 \leqslant z'_1 \leqslant \ldots \leqslant z'_k < a$ for each $k \in {\mathbb{Z}}_{\geqslant 0}$) or decreasing (i.e. $z'_0 \geqslant z'_1 \geqslant \ldots \geqslant z'_k > a$ for each $k \in {\mathbb{Z}}_{\geqslant 0}$).

\[Convergence from below and above\] \[def:convergence\_from\_below\_and\_above\] Let $\{z_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ be a convergent sequence from ${\mathscr{Z}}$. If $\{z_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ converges to $p \in \overline{{\mathscr{Z}}}$, then we write $z_i \to p$.

-   We say that $z_i \to p$ [*from below*]{} if there is a $\mu \in S^1 \setminus \{p\}$ such that $z_i \in [\mu, p]$ from some step.

-   We say that $z_i \to p$ [*from above*]{} if there is a $\nu \in S^1 \setminus \{p\}$ 