Problem #PRU-5078

Problem

Theorem: All people have the same eye color.

"Proof" by induction: This is clearly true for one person.

Now, assume we have a finite set of people, denote them as $a_1,a_2,...,a_n$, and the inductive hypothesis is true for all smaller sets. Then if we leave aside the person $a_1$, everyone else $a_2,a_3,...,a_n$ has the same color of eyes and if we leave aside $a_n$, then all $a_1,a_2,a_3,...,a_{n-1}$ also have the same color of eyes. Thus any $n$ people have the same color of eyes.
Find a mistake in this "proof".