We consider a linear ill-posed operator equationAx = yin Hilbert spaces. An algorithmRε:Y→Xfor solving this equation with given inexact right-hand sideyε, such that, is called order optimal if it provides best possible error estimates under the assumption that the minimal norm solutionx*of this operator equation fulfils some smoothness condition. It is shown that if such an algorithm is slightly modified tothen it is a regularization method, i.e., we havewithout additional conditions onx*.