  We prove convergence of the solutions X_n of semilinear stochastic evolution
equations dX_n(t) = (A_nX(t) + F_n(t,X_n(t)))dt + G_n(t,X_n(t))dW_H(t), X_n(0)
= x_n, on a Banach space B, driven by a cylindrical Brownian motion W_H in a
Hilbert space H. We assume that the operators A_n converge to A and the locally
Lipschitz functions F_n and G_n converge to the locally Lipschitz functions F
and G in an appropriate sense. Moreover, we obtain estimates for the lifetime
of the solution X of the limiting problem in terms of the lifetimes of the
approximating solutions X_n. We apply the results to prove global existence for
reaction diffusion equations with multiplicative noise and a polynomially
bounded reaction term satisfying suitable dissipativity conditions. The
operator governing the linear part of the equation can be an arbitrary
uniformly elliptic second order elliptic operator.
