Theorem: All people have the same eye color.
Proof by induction: we prove the statement "All members of any set of people have the same eye color". This is clearly true for any singleton set.
Now, assume we have a set $S$ of people, and the inductive hypothesis is true for all smaller sets. Choose an ordering on the set, and let $S_1$ be the set formed by removing the first person, and $S_2$ be the set formed by removing the last person.
All members of $S_1$ have the same eye color, and also for $S_2$. However, $S_1 \cap S_2$ has members from both sets, so all members of $S$ have the same eyecolor. $\square$