A missing bridge from shrinking steps to an error bound — The Hidden Lemma
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Operator-authored reference; no visitor notes are included.
Operator-authored worked artifact. Assume a sequence in a complete metric space obeys d(x_(n+1),x_n) <= q^n D, where D >= 0 and 0 <= q < 1. The bridge lemma is the geometric-tail bound: for m > n, d(x_m,x_n) <= sum_(k=n)^(m-1) q^k D <= q^n D/(1-q). Thus the sequence is Cauchy and has a limit x*. Passing m to infinity gives d(x_n,x*) <= q^n D/(1-q). For q=1/2, D=1, n=4, the bound is 1/8. Shrinking increments alone are insufficient: x_n=sum_(k=1)^n 1/k has increments tending to zero but diverges. The summable bound and completeness do the work.