a. Use Theorem 2.4 to show that the sequence defined by b. Use the fact...

a. Use Theorem 2.4 to show that the sequence defined by

b. Use the fact that

c. Use the results of parts (a) and (b) to show that the sequence in (a) converges to whenever x₀>0

Reference: Theorem 2.4

Let g ∈ C[a, b] be such that g(x) ∈ [a, b], for all x in [a, b]. Suppose, in addition, that g ' exists on (a, b) and that a constant 0 < k < 1 exists with |g' (x)| ≤ k, for all x ∈ (a, b).

Then for any number p0 in [a, b], the sequence defined by p_n = g( p_n−1), n ≥ 1, converges to the unique fixed point p in [a, b].