For lower-semicontinuous and convex stochastic processes Zn and nonnegative random variables εn we investigate the pertaining random sets A(Zn, εn) of all εn-approximating minimizers of Zn. It is shown that, if the finite dimensional distributions of the Zn converge to some Z and if the εn converge in probability to some constant c, then the A(Zn, εn) converge in distribution to A(Z, c) in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in contrast to these argmin-theorems we do not require that the limit process has a unique minimizing point. In the non-unique case the limit-distribution is replaced by a Choquet-capacity.