We consider the the problem of approximating a given object x (say, a function) by a sequence (x_n), whose terms belong to the prescribed family of sets (A_n)$ (for instance, A_n may be
the space of polynomials of degree less than n). For each n, compute the distance E_n from x to A_n. How does the sequence (E_n) behave? What are the connections between its rate of
decrease and the properties of x? Can we discern any patterns in the sequence (E_n)? We attempt to answer these questions for different families (A_n).