  We give a brief account on a basic result (Lemma \ref{lem2}) which is a very
useful tool in proving various convergence theorems in the framework of the
iterative approximation of fixed points of demicontractive mappings in Hilbert
spaces. This Lemma relates the class of quasi-nonexpansive mappings, by one
hand, and the class of $k$-demicontractive mappings (quasi $k$-strict
pseudocontractions), on the other hand and essentially states that the class of
demicontractive mappings, which strictly includes the class of
quasi-nonexpansive mappings, can be embedded in the later by means of an
averaged perturbation. From the point of view of the fixed point problem, this
means that any convergence result for Krasnoselskij-Mann iterative algorithms
in the class of $k$-demicontractive mappings can be derived from its
corresponding counterpart from quasi-nonexpansive mappings.
