Discrete Grönwall inequality

Statement

Let $(u_k)_{k \ge 0}$ be a sequence of real numbers with $u_0 \ge 0$, and let $(b_k)_{k \ge 0}$ and $(c_k)_{k \ge 0}$ be sequences of non-negative real numbers. Assume that $u_{k+1} \le (1 + c_k)\, u_k + b_k$ for each $k \ge 0$. Then for each $n \ge 0$, $$u_n \le \left( u_0 + \sum_{k=0}^{n-1} b_k \right) \exp\left( \sum_{k=0}^{n-1} c_k \right).$$