MATH.21:4 - Solution

Local mantra: name the limiting object and its next use; construct the approximation family; choose and establish convergence; obtain the object in the admitted space; carry only the justified properties and operations; return a sufficient finite result or the remaining mathematical question.

MATH.21:4.1 - State what approximation must preserve

Name the space of intended objects, the approximating objects and how the latter are compared with the former. If they initially have different kinds, supply an embedding or interpretation. For example, a rational number can approximate a real number; a finite word can determine the beginning of an infinite word.

State the requested result. Does the receiver need a value, an operation on the limit, an existence theorem, or a finite observation? For a numerical use, specify the error quantity and tolerance. For a prefix use, specify which entries must be settled. The mathematical question determines the approximation requirement.

Choose a convergence notion that supports that result. In a metric space, with distance d, a sequence x_n converges to x when, for every positive epsilon, all sufficiently late terms satisfy d(x_n,x)<epsilon. For functions, pointwise convergence allows the sufficiently late index to depend on the input; uniform convergence requires one index to work for all inputs in the named set. For infinite words, convergence can mean eventual agreement on every fixed finite prefix.

These are different constructions of the convergence requirement. Use their definitions to decide what they permit. Numerical distance is useful when it measures the requested difference; finite observation or another mathematical relation can be more suitable elsewhere.

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.

MATH.21:4.3 - Obtain the limit where it is needed

If a candidate object is already available, prove convergence to it by the selected definition or a suitable theorem. Otherwise use an existence result whose conditions the approximation satisfies.

For example, completeness of a metric space means that every Cauchy sequence in it has a limit there. Establishing a Cauchy condition and applying a known completeness result can obtain an object without guessing its closed form. A local existence argument may suffice even when the whole space is incomplete.

When the limit falls outside the original objects, decide whether the receiving work admits an extension. Construct the new objects and the embedding of the old ones; prove the properties the subsequent work will use. One route takes suitable Cauchy sequences as representations and identifies those whose mutual distance tends to zero. MATH.2 supplies the quotient operation: operations on equivalent representatives must produce equivalent outputs, so the quotient result is independent of the representative. The real-number construction is one instance of this route.

Establish uniqueness when the receiver needs a single result. In a metric space, if x_n tends to both x and y, the triangle inequality gives d(x,y)≤d(x,x_n)+d(x_n,y); the right side can be made arbitrarily small, so x=y. Other convergence structures need their own uniqueness condition or an explicitly retained class of possible limits.

If convergence or existence fails, return the failed condition and the still valid finite information. That can suggest a different space, a different approximation family, or a weaker requested conclusion.

MATH.21:4.4 - Pass properties and operations through the limit

For every operation used next, establish the relevant interchange. If F is continuous for the chosen source and target convergence, x_n→x gives F(x_n)→F(x). The argument can be local to this family; global continuity is sufficient in many cases but is more than every use needs.

Check the property actually consumed. A limit of objects in a closed subset remains in that subset. Other properties can disappear: finite words padded with zeros can converge to an infinite word with infinitely many ones.

For a function limit, distinguish a claim about its values from a claim about integration or differentiation. The required limit theorem may ask for uniform control, domination, or another hypothesis specific to the operation. In :5.3 a uniform value bound supports integration over a finite interval, but it does not support differentiating the approximations to obtain the limiting derivative.

When an interchange fails, retain the established limit and repair the affected operation. Strengthen the convergence condition, choose another family, restrict the domain, or calculate that operation by a different argument. Select the repair from the receiver’s need.

MATH.21:4.5 - Obtain a sufficient finite result

Translate the receiving request into a condition on a stage. If an established bound gives d(x_n,x)≤e_n, a stage with e_n within the requested tolerance supplies the approximation. For nested numerical intervals, the midpoint has error at most half their width. For compatible prefixes, a stage containing all requested entries supplies them.

When execution must find that stage, provide either an obtaining rule or a recognizable stopping condition with a reason it will be reached. An explicit rate of convergence is one option. An enclosure whose width can be checked during refinement may suffice without a rate fixed in advance.

A residual is useful only through its established relation to the requested error. A small equation residual can accompany a large error in the unknown; the comparison needed by the use must connect them.

Return the approximation with the mathematical bound or settled observation it supplies. If only existence has been established, retain that result and identify the missing obtaining procedure for a computational continuation. Numerical conditioning, finite arithmetic and execution cost belong to that continuation.

MATH.21:4.6 - Use and revise the construction

Use the limiting object in the argument, or use the finite result for the selected observation. Keep the convergence notion and any condition needed by the next operation with that use.

A changed tolerance can require a later stage. A changed operation can require a stronger convergence result. A changed space can remove existence or uniqueness. Reopen the affected argument and retain the finite constructions and proved consequences that survive.

In modeling, interpret the resulting mathematical consequence against the subject using C.29 and the corresponding modeling method. The approximation’s mathematical convergence and its ability to answer the subject question remain separately assessable.