MATH.21:4.2 - Construct and compare the finite stages

Give the approximating rule and establish the relations on which improvement depends. For nested intervals, prove containment and decreasing width. For compatible prefixes, prove that later stages retain the earlier entries. For a series or iterative construction, obtain a bound on the unresolved remainder.

The same argument must cover the stages used by the conclusion. A finite sample can suggest a bound or expose a failure; a claimed statement about all later stages needs the corresponding argument.

When the candidate limit is not yet available, a Cauchy condition can express progress using only the approximations. In a metric space it says: for every positive epsilon there is N such that d(x_m,x_n)<epsilon whenever m and n are at least N. This compares the entire remaining tail.

If a stopping rule uses only successive changes, derive how they bound the remaining tail. Successive changes of size 1/n become small, yet their accumulated sum is unbounded. A summable bound, a contraction estimate or another proved remainder relation can make a successive-change test useful.

Retain a cheaper construction when it already supplies the requested observation. Refinement need not improve every property at every stage.