Suppose the last moment the car was at rest is at $t_0=0$. What time is the first moment the car moves?

Suppose you pick some number $t_{first} > 0$. It doesn't matter what number you pick. You can always show that there is a problem with it.

There is a number $t_{first}/2$. This number is bigger than $0$, so it is after $t_0$. So the car is not at rest. It is before $t_{first}$, so the car has not started moving yet.

This is a contradiction. It shows you cannot pick a moment after $t_0$ that is the moment the car first moves. You must conclude the car starts moving at $t_0$. So you are right to be suspicious of his answer.

Suppose the last moment the car was at rest is at $t_0=0$. What time is the first moment the car moves?

Suppose you pick some number $t_{first} > 0$. It doesn't matter what number you pick. You can always show that there is a problem with it.

There is a number $t_{first}/2$. This number is bigger than $0$, so it is after $t_0$. So the car is not at rest. It is before $t_{first}$, so the car has not started moving yet.

This is a contradiction. It shows you cannot pick a moment after $t_0$ that is the moment the car first moves. You must conclude the car starts moving at $t_0$. So you are right to be suspicious of his answer.

This question is continuing to generate interest, so I thought I would add to my answer.

First numbers make a difference. This paradox was first proposed by Zeno. It wasn't settled for thousands of years. But it is quite straightforward to modern mathematicians.

That said, you still have to be careful with it. I haven't been careful enough. The resolution I proposed is that at $t = t_0$, the car is both at rest and has started moving. This sounds like a paradox.

That's OK - I'm a physicist. The purpose of physics is to model the behavior of the universe. A logical inconsistency this small doesn't change the predicted behavior of the car at all. Physicists make bigger approximations all the time.

But it isn't good enough for a mathematician. Mathematicians model ideas. Logic is their only tool. If the logic isn't airtight, a false statement can be proven. And one false statement can be used to prove more. The entire structure of mathematics would collapse.

So a mathematician would pay more careful attention to definitions. What exactly is meant by "at rest", or "has started moving"? Is the car at rest if its position hasn't changed? If its velocity is $0$?

He would note that I haven't really proven that the car has started moving at $t_0$. Just at all times after $t_0$. There is no such thing as the first moment where the car's position has changed. There is only a last moment where it has not.

This kind of distinction is crucial to point set topology. It determines many interesting properties of topological spaces. (Where but CalTech would an introduction like the link come from the Division of Humanities and Social Studies?)