Our definition of separation is:
If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have that $\{x_n\} \in A$, $x_n \to x$, with $x \in B$ (and also vice versa).
Considering the $2$ sets $\{(x,0) \in \mathbb{R^2} : x \in \mathbb{R}\}$ and $\{(x,\frac{1}{x}) : x >0\}$ (assuming they are $A$ and $B$ as in the definition, and the union of them is $C \subset \mathbb{R}$), the shortest distance between them is $0$, which seems like it would imply that they "meet" at some point, because the distance between them is possible to be $0$, as $x \to \infty$. I don't have any intuition about sequences for this, so how can I tell if these two sets are separated or not?
HINT: Show that every point $\langle x,0\rangle$ of $A$ has an $\epsilon$-ball around it that is disjoint from $B$. In that case can any sequence in $B$ converge to $\langle x,0\rangle$?