2.1 Monotonic Subsequences

In reverse order to how it was discussed in tutorials, we begin with some special types of sequences. First, we need to define what these sequences are. This definition might look long, but it’s really just five concepts grouped together.
Definition 2.1: (Monotonic Sequences)

Let \((a_n)_{n\in\mathbb{N}}\) be a real sequence. Then if \(\forall n \in \mathbb{N}\):

  • \(a_{n+1} \geq a_n,\;\) \((a_n)_{n\in\mathbb{N}}\) is increasing,
  • \(a_{n+1} > a_n,\;\) \((a_n)_{n\in\mathbb{N}}\) is strictly increasing,
  • \(a_{n+1} \leq a_n,\;\) \((a_n)_{n\in\mathbb{N}}\) is decreasing,
  • \(a_{n+1} < a_n,\;\) \((a_n)_{n\in\mathbb{N}}\) is strictly decreasing.
If the sequence has any one of these properties, it is called monotone, or a monotonic sequence.

For a given sequence \((a_n)\), the two main ways of checking monotonicity are by considering \(a_{n+1} - a_n\) and/or \(\frac{a_{n+1}}{a_n}\), and comparing these objects to \(0\) and \(1\) respectively. The second of these methods is especially useful when you’re dealing with powers of \(n\), such as for the sequence \((b_n)\) in Exercise Sheet 6, Homework Question 2.

A useful theorem for these sequences is the following1:
Theorem 2.1:

Let \((a_n)_{n\in\mathbb{N}}\) be a bounded, monotone sequence. Then \((a_n)_{n\in\mathbb{N}}\) is convergent.

In fact, if a sequence \((a_n)_{n \in \mathbb{N}}\) is increasing, then it converges to the supremum of the set of \(a_n\) values, and if it is decreasing, then it converges to the infimum of the set of \(a_n\) values.

  1. If you’re interested, this statement is completely equivalent to the completeness axiom from Section 2 of the lecture notes. In fact, back in the prehistoric times of 2016 — when I took the course — this result was stated as the completeness axiom.↩︎