We assume that the Wightman distributions are tempered and that they are boundary values of functions analytic in the forward tube. Under these conditions a sequence of such distributions may converge in Borchers' topology. The necessary and sufficient conditions for such convergence are spelled out in terms of the corresponding analytic functions. The two cases with and without the assumption of the spectral condition are separately treated. A discussion of other topologies and some examples of the use of this technique are given.