In this article, the authors exploit a close connection between the fully nonlinear equation \[ \min\{L_1 u, L_2 u\} = 0 \text{ in } \Omega, \;u= g \text{ on } \partial \Omega,\tag{1} \] and the obstacle problems for \(L_1\) and \(L_2\) respectively. Here, \(L_1, L_2\) are operators in a suitable class (e.g., they could be such that maximum principles are valid and continuity conditions hold for the coefficients). More precisely, the authors show that it is possible to construct solutions to (1) by an iterative procedure: Starting from a solution to \(L_1 u_1= 0\) in \(\Omega\), \(u_1=g\) on \(\partial \Omega\), for \(n\geq 2\) the authors define \(u_n\) as the solution to the obstacle problem for \(L_1\) if \(n\) is odd, and for \(L_2\) if \(n\) is even, with the constraint that \(u_n \geq u_{n-1}\). They show that in the limit \(n\rightarrow \infty\) this yields a viscosity solution to the fully nonlinear problem (1). Furthermore, various extensions, e.g., to parabolic operators and uncountable index sets, are discussed.