We provide a bounded backstepping result that ensures global asymptotic convergence for a broad class of partially linear systems with an arbitrarily large number of integrators. We use one artificial delay, and we assume that the nonlinear subsystems satisfy a converging-input-converging-state assumption. When the nonlinear subsystem is control affine with the state of the first integrator as the control, we provide sufficient conditions for our converging-input-converging-state assumption to hold. Our example illustrates the novelty and utility of our main result.