Question

Define a sequence from R as follows. Fix r > 1. Let a1 = 1 and define recursively, an+1 = (1/r) (an + r + 1). Show, by induction, that (an) is increasing and bounded above by (r+1)/(r−1) . Does the sequence converge?

Answer 1

Monotone Convergence Theorem: A monotone sequence of real numbers is convergent if and only if it is bounded.

Theorem: If a sequence is monotone and bounded, then it converges.

Proof. We consider two cases.
Case (i) (if a_n is increasing). Since (an) is bounded, implies that the set A = {a_{​​​​​​n} | n ∈ N} is a bounded subset of R. Hence the Least Upper Bound Property of R implies that A has a supremum. Let s = sup A. We will show that (a_{​​​​​​n}) → s. Given ε > 0, since s = sup A, there exists some element a_N ∈ A such that s ε < a_{​​​​​​N} .
Given any n ≥ N, since (a_{​​​​​​n}) is increasing, we have that a_{​​​​​​N} ≤ a_{​​​​​​n}, and therefore s − ε < a_{​​​​​​n} as well. But
since an ∈ A, it follows that a_{​​​​​​n} ≤ s < s + ε. Therefore s − ε < an < s + ε and hence |an − s| < ε for all
n ≥ N. Hence we have shown that for all ε > 0, there exists N ∈ N(set of natural numbers) such that if n ≥ N, then |an − s| < ε, which implies that (a_n) → s. Hence indeed (a_{​​​​​​n}) converges.
Case (ii) (if a_n is decreasing). Similer as case (i)