There's this classic proof by induction that is FALSE:

The claim is that cars are all the same colour. This is equivalent to the claim that any set of cars

must contain cars of the same colour. The claim is trivially true for the base case of n = 1 cars

since, after all, a car has only one colour. Next, we’ll assume that the claim is true for all sets of

n cars are we’ll now consider a set of n + 1 cars, {car 1, car 2, ..., car n, car n + 1}. We’ll consider

two subsets of this set C1 = {car 1, car 2, ..., car n} and C2 = {car 2, ..., car n, car n + 1}, each of

which is a set of n cars. Therefore, by our induction hypothesis, C1 and C2 only contain cars

of a single colour. But then again, since there’s overlap in the entries of C1 and C2 the colour

of each must be the same. Therefore {car 1, car 2, ..., car n, car n + 1} must have only cars of a

single colour. Thus, cars are all of the same colour

Apparently to suggest that n = 1 implies n = 2 is false (therefore proving the proof as false) and I want to more clearly understand why that would be the case. Does it have to do with how C1 and C2 are related to each other? Perhaps you can't suggest that car n in C1 is the same colour as car n in C2?