Without loss of generality, let us label the gas stations , where holds the most amount of gas holds the second most amount, and so on. I'm only focusing on the 'worst' case as well, in other words, that all the gas stations hold exactly the required amount to go around the track once.

We are given that

We know that the base case, n=1, is true. If there is one gas station on the track, then we place the car right at the gas station.

Then we can invoke the induction step that if we have n gas stations, then we can place the car somewhere on the track and it will go around the track once.

Now we have to prove P(n+1), that is to say that if we have n+1 gas stations then we have to prove that we can place the car somewhere on the track and it will go around the track once. careful, however.

For this step we can only assume

(S1)not

However, we do know that by (S1) . However, since we ordered the gas stations we know that " holds the least amount of gas amongst all the other gas stations." (S2)

and from here I don't know how to be more rigorous to prove that this is enough to show that the car can go around the track exactly once... but at least it's a start.

EDIT: I want to say something like "No matter where you put gas station , the car will reach with the amount of gas that it has stored on its trip." with proof invoking (S1), (S2) and our induction step assumption.