The powers of a function f: A → A are defined recursively bySuppose un and vn are sequence...

The powers of a function f: A → A are defined recursively by

Suppose un and vn are sequences defined recursively by

and, for n ≥ 1,

(a) Prove that for n ≥ 1.

(b) Prove that un is an increasing sequence; that is, un+1 > un for all n ≥ 1.

(c) Prove that vn is a decreasing sequence; that is, vn+1 < vn for all n ≥ 1.

(d) Prove that for all n ≥ 1.

(This problem is taken from a Portuguese examination designed to test the level of mathematical knowledge of graduating high school students. It was reprinted in Focus, the newsletter of the Mathematical Association of America 13, no. 3, June 1993, p. 13.)