A generalization of successive approximations to the square root 
of a positive operator on a Hilbert space due to Riesz-Nagy and Halmos is discussed 
from the viewpoint of operator means and operator inequalities: 
For the arithmetic mean $\nabla $ and a positive operator $A$, 
a sequence $\{ A_n \}$ satisfying 
   \[
 A\geq A_n^2\geq 0 \quad \mbox{and} \quad 
 A_{n+1}\geq 2A_n \ \nabla (A-A_n^2) \qquad \mbox{for $n=0,1,2,\cdots $}
 \]
 converges monotone increasingly to the square root $\sqrt{A}$ of $A$ 
in the strong operator topology, in which the operator sequence 
is selected seemingly at random. Moreover, we discuss a harmonic mean version 
for Newton's method.