Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$

I would like to know if the sequence $(u_n)_{n \geq 1}$ defined by $$\forall n \in \mathbb{N},~u_n = \sum_{i=1}^n \gamma_i \prod_{j=i+1}^n (1-\gamma_j)^2$$ is bounded

**Note**

I have proven that with the same conditions on $(\gamma_n)_{n \geq 1}$, we have $$ \sum_{i=1}^{t} \gamma_i^2 \prod_{j = i+1}^{t} (1-\gamma_j)^2 \underset{n \to \infty}{\longrightarrow} 0$$ using the fact that $\prod_{j = i+1}^{t} (1-\gamma_j)^2 = e^{2 \sum_{j = i+1}^{t} \ln(1-\gamma_j)} \leq e^{- 2 \sum_{j = i+1}^{t} \gamma_j}$