*Numerical* Convergence of the Babylonian Method? I understand the sequence $x_{n+1} = \frac12\left(x_n + \frac2 {x_n}\right) $ converges to $ \sqrt2 $ algebraically.
That is proved by means of fixed-point method or monotone convergence theorem and so on.
But, I do want to know whether the sequence converges numerically (and if so, the proof of it).
So, here is my question. 
Does the following algorithm always terminate?

  
*
  
*Choose an initial number  $ x_0 > 0 $.
  
*Calculate $x_{n+1}=\frac12\left(x_n + \frac2 {x_n}\right)$ to one more places of decimals you need.
  
*Truncate the number $x_{n+1}$ to places of decimals you need. 
  
*When $x_{n+1}=x_n$ occurs, this algorithm terminates.
  

In step 3, you need to truncate the number, and I'm worried about the effect of truncation on the convergence.
 A: Actually, this issue does come up in square root algorithms if you are not careful. To take a simple example, suppose we work in base $12$ and we truncate using the floor function. Start with a bigger approximation to $\,\sqrt{2}\,$ 
which is
 $\, 17/12. \,$ Using the Babylonian algorithm, the next approximation is
 $\, (17/12 + 24/17)/2 = 577/408 = 16.9705.../12 \,$ and if you truncate this to
 $\, 16/12, \,$ then the next approximation is $\, (16/12 + 24/16)/2 = 17/12 \,$ and we have a loop, unless we terminate the algorithm at this step since we are getting a  bigger approximation. The exact behavior of the sequence of approximations to $\, \sqrt{x} \,$ depends on the exact details of how the arithmetic is done, but except for the first iteration, you should always terminate if this sequence stops decreasing.
This issue is a very general one. Suppose you have $\,\{x_n\},\,$ a sequence of real numbers given by a recurrence $\, x_{n+1} = f(x_n) \,$ that converges to a finite limit. Now replace the real numbers by $\,F,\,$ a finite set of numbers such as used in a computer system, fixed point or floating point. Then the computing function $\, f_F : F \to F  \,$ corresponding to $\,f,\,$ no matter what the initial value or even if the original $\,f\,$ converges or not, must eventually loop, and in best cases the loop is a fixed point.
Just as a curiousity, this kind of loop happens in other bases, namely,
 $\, \{2, 12, 70, 408, 2378, \dots\} \,$ which is OEIS sequence A001542 which relates them to continued fraction convergents to $\, \sqrt{2}. \,$
