Understanding growth function of closed intervals in $\mathbb{R}$
Let the real numbers be $r_1 &lt; \cdots &lt; r_m$. The intersection $H \cap C$ could be of the form $\{r_i,\ldots,r_j\}$ for $1 \leq i \leq j \leq m$, or empty. There are $\binom{m+1}{2}$ of the first type, and $1$ of the second type.
For example, if $m = 1$ then the possible intersections are
$$ \emptyset, \{r_1\}, $$
if $m = 2$ then the possible intersections are
$$ \emptyset, \{r_1\}, \{r_2\}, \{r_1,r_2\}, $$
and if $m = 3$ then the possible intersections are
$$ \emptyset, \{r_1\}, \{r_2\}, \{r_3\}, \{r_1,r_2\}, \{r_2,r_3\}, \{r_1,r_2,r_3\}. $$