This paper studies the backward stochastic differential equation (BSDE) of the form
where is a Brownian motion and is a non-anticipative Lipschitz-continuous function. As usual, a process is called a supersolution of the BSDE if it is of the above form for some adapted right-continuous, increasing process and some predictable, square-integrable process . The main result of the paper is a theorem which asserts that, under suitable integrability conditions, the pointwise monotone limit of a sequence of supersolutions is again a supersolution of the BSDE. Moreover, it is shown that the corresponding integrands converge weakly in and strongly in each with ; the processes converge weakly in . As an application of this result, the author proves a generalization of the classical Doob-Meyer decomposition to so-called -supermartingales. These processes refer to a suitably defined nonlinear expectation operator in essentially the same way as usual supermartingales to usual expectations. As a second application, the author shows that there exists a minimal supersolution of the BSDE which is subject to fairly general state- and time-dependent constraints.