MATH.21:2 - Problem

Finite stages can look increasingly satisfactory while leaving the mathematical result undetermined. The intended limit may be absent from the chosen space. Agreement at each fixed input may fail to control all inputs together. An operation valid on every stage may cease to be valid at the limit.

Even established convergence can leave a practical question open: which stage supplies the requested result? Conversely, a finite observation can be available before an entire limiting construction has been resolved. The method must connect the convergence claim to the particular subsequent use.