The influence of the fluctuations of the electromagnetic environment on resonant tunneling is investigated in a rigorous way. Resonance-level broadening is found to be governed by the environmental impedance and temperature. It is found that the high-frequency limit of resonant tunneling devices is determined by the discharging time of the whole circuit. The obtained criterion for the observation of single-electron charging effects in resonant tunneling Rg8${\mathit{R}}_{\mathit{Q}}$=8\ensuremath{\pi}\ensuremath{\Elzxh}/${\mathit{e}}^{2}$ is much stronger than the Rg${\mathit{R}}_{\mathit{Q}}$ used in the theory of the Coulomb blockades.