BG-MO 2020 Problem 2

Let $b_1$ , $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$ , $a_1$ , $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$ , $\dots$ , $n$ . Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ Click to expandI believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold