# Understanding growth function of closed intervals in $\mathbb{R}$

I as studying VCdimensions and growth functions and found the following example on Wikipedia:

The domain is the real like $$\mathbb{R}$$. The set H contains all the real intervals, i.e., all sets of form $$\{c \in [x_1, x_2] | x \in \mathbb{R}\}$$ for some $$x_{0, 1} \in \mathbb{R}$$.

For any set C of m real numbers, the intersection $$H \cap C$$ contains all runs of between 0 and m consecutive elements of C. The number of such runs of $${m+1 \choose 2} + 1$$, so Growth(H, m) = $${m+1 \choose 2} + 1$$.

Can anyone please explain to me what does the term "all runs of between 0 and m" refer to here and why the growth function is $${m+1 \choose 2} + 1$$ and not $${m+1 \choose 2}$$?

Thank you very much!