We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a quasi-variational inequality and it is $C^1$ across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman and does not use the \emph{the theory of vicosity solutions}, since our proof relies on constructing a sequence of functions, each of which is a value function of an optimal stopping time for a \emph{diffusion}. (Also, the previous proof holds only for a certain range of parameters.) The sequence is constructed by iterating a functional operator that maps a certain class of convex functions to smooth functions satisfying variational inequalities (or to value functions of optimal stopping problems for a geometric Brownian motion). The approximating sequence converges to the value function exponentially fast, therefore it constitutes a good approximation scheme, since the optimal stopping problems for diffusions can be readily solved using the well-know numerical schemes such as SOR.