I'm reading about the proof of Helly's Intersection Theorem:
I'm unable to understand the reasoning behind the argument
If $n=2,$ the set $C_{n} \cup C_{-n}$ is convex, so by the Mean Value Theorem it must contain a point in $H$...
Could you please elaborate on how we obtain the contradiction? Thank you so much!
It is not the mean value theorem that is being used, it is the intermediate value theorem.
Let $x_1 \in C_n$ and $x_2 \in C_{-n}$. Since $C_n \cup C_{-n}$ is convex, we have a path $p:[0,1] \to C_n \cup C_{-n}$ connecting both. The function $f \circ p$, where $f(x)=\langle x , y\rangle$, is then a real continuous function which is greater than $a$ at $1$ and smaller than $a$ at $0$. By the intermediate value theorem, there is a point $t_0$ such that $(f \circ p)(t_0)=a$, i.e., $\langle p(t_0), y \rangle = a$. This point $p(t_0)$ is by definition in $H$.