Before revision:  With the bounded terminal condition as well as an additional local Lipschitz continuity, we give a simple and streamlined proof for the existence and the uniqueness of the solution . The universal bound and the stability result for the locally Lipschitz BSDEs with coefficients in the BMO space enable us to show the strong convergence of a sequence of globally Lipschitz BSDEs .
After revision:  With the bounded terminal condition as well as an additional local Lipschitz continuity, we give a simple and streamlined proof for the existence and the uniqueness of the solution without using the comparison principle. The properties of locally Lipschitz BSDEs with coefficients in the BMO space enable us to show the strong convergence of a sequence of globally Lipschitz BSDEs .