how to estimate a multiple indefinite integral with constant integrand?

Question

Let $\rho_i,\eta_i\in[a,b]$. How to prove the following inequality? \begin{equation*} \left|\int_{\eta_{k+1}}^{\rho_{k+1}}d\rho_k\int_{\eta_{k}}^{\rho_{k}}d\rho_{k-1}\cdots\int_{\eta_{2}}^{\rho_2}d\rho_1\right|\leq \frac{(b-a)^k}{k!}. \end{equation*} I have spent much time, but nothing come out. I think it could be probably proven by Induction, but I didn't get a right way.