Proving the pigeonhole principle in Cutting Planes
The proof is by induction. Here is the inductive step.

Suppose that we know both of the following inequalities:
$$
x_1 + \cdots + x_m \leq 1 \\
x_2 + \cdots + x_{m+1} \leq 1
$$
Add them to get
$$
x_1 + 2x_2 + \cdots + 2x_m + x_{m+1} \leq 2
$$
Add the axiom $x_1 + x_{m+1} \leq 1$ to get
$$
2x_1 + \cdots + 2x_{m+1} \leq 3
$$
Divide by 2 and round down to get
$$
x_1 + \cdots + x_{m+1} \leq 1
$$
I'll let you fill in the remaining details.