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.