# Prove by induction that if the first car stops, then all cars will stop

$$n$$ cars are travelling down a narrow one-way street. We know that:

• The distance $$d$$ between each two cars is the same.
• The safe breaking distance $$b$$ is the minimum distance between two cars that is needed for the second car to stop on time if the car in front suddenly breaks.
• $$d < b$$

Prove by induction or refute: if the first car suddenly stops moving, all cars will stop moving. Before you do the induction state the property $$P$$ you are using in the induction axiom.

What I came up so far:

Proof:

Base case: $$n=1$$

if there is only one car traveling down a narrow one way street, than obviously only 1 car will stop moving -> therefore the assertion is true

Induction step: $$n= k+1$$

I am stuck here---- need a hint

It's unknown what will happen in the case $n\gt1$. As the cars don't have sufficient distance to break safely, the second car may crash into the first car. The given information doesn't determine what happens in this case; for instance, the brakes (not breaks) of both cars might break and the cars might skid along the (possibly inclined) street indefinitely.