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.