Practical implementation of the various regularization methods for unstable problems which are proposed in the articles by N.~A. Tikhonov, M.~M. Lavrent'ev and others involves constructions of some finite-dimensional analogs of the regularization methods. Therefore, the problem arises of studying convergence of solutions to these approximation problems. Mainly, the results in this direction yield sufficient conditions for convergence in the case of bounded operators. In the article under consideration, the authors obtain necessary and sufficient conditions for convergence of approximations for closed operators. The criteria for convergence of approximations are exposed in the general case for an arbitrary approximation scheme as well as in one particular case.