The authors present a new proof of well-posedness of an infinite-dimensional stochastic evolution equation expressed in variational form \[ dX(t) = A(t,X(t))dt + B(t,X(t)dW(t), \] where \(A\) is a random time-dependent operator defined on a Banach space into its dual space, \(W\) is a cylindrical Wiener process, and \(B\) is a random time-dependent map taking values in a space of Hilbert-Schmidt operators. The authors eschew the classical approach, based on finite-dimensional projections, in favor of a seemingly more sophisticated, but technically simpler, non-linear approximation argument applied in the infinite-dimensional setting.