Consider the following sum: $$ S = \sum_{i = 1}^n x_{i + 1} \cdot (x_{i + 1} - x_i)^2 \cdot \prod_{j = 2}^i (1 - (x_j - x_{j - 1})^2) $$ where $ 0 \leq x_i \leq 1$ and $x_i \leq x_{i + 1}$ for any $i$.
I am looking for an upper bound on this sum $S$, as tight as possible, preferably. I tried to compute exactly this sum and I couldn't obtain any nice expression. Then, I tried to upper bound using AM–GM inequality or using the inequality $y^2 \leq y$ for $y \in [0, 1]$, but either the resulting expression was very hard to compute or the final bound was very weak.
If anyone has any suggestion on what inequality could be suitable, I would highly appreciate it!