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.