Consider a chain of binary relations $\{a_i\}_{i \in \mathbb{N}}$, where $\forall i \in \mathbb{N} : a_i \subseteq X \times X$, where $X$ is a countable set.
Additionally the following holds:
$$ \forall i \in \mathbb{N} . a_i \subseteq a_{i+1} $$
Is it the case that for all $(x, y) \in \bigcup_{i \in \mathbb{N}} a_i$ there exists an index $i(x, y)$ such that $(x, y) \in a_{i(x, y)}$?
Note that the index depends on $(x, y)$; it surely is not the case that there exists some index $i'$ such that $\forall (x, y) \in \bigcup_{i \in \mathbb{N}} (a_i) . (x, y) \in a_ {i'}$