algorithms – Gas Station problem proof

I’m trying to prove the solution for the problem Gas Station (LeetCode 134)

Let’s define $d_i$ to be the difference $g_i – c_i$.

Basically it’s a greedy solution where you look for the first $k$ s.t.

$$sum_{i=k}^n d_i ge 0$$

Let’s denote the following:

$$A = sum_{i=1}^k d_i$$

$$B = sum_{i=k+1}^n d_i$$

Notice that by definition of $k$, $A<0$ and $Bge 0$. Since we know that there’s a unique solution

$$A+B ge 0 implies Bge -A$$

We want to show that $B-A ge 0$. This will show that we have enough gas for $A$ after finishing the $B$ part.

So $$B-A ge B+B ge 0$$

Is my proof correct?

Thanks