A sequence \(\{a_n\}\) of real numbers is called increasing (some authors use the term nondecreasing) if \(a_n \leq a_{n+1}\) for all \(n\). It is called strictly increasing if \(a_n < a_{n+1}\) for all \(n\). The sequence is called decreasing if \(a_n \geq a_{n+1}\) for all \(n\), etc.
A sequence is called monotonic (or a monotone sequence) if it is either increasing (strictly increasing) or decreasing (strictly decreasing).
Example Classify each of the following sequences as increasing, decreasing, or neither.
(a) \(a^n\) is increasing if \(a>1\) and is decreasing if \(0<a<1\).
(b) \(n(-1)^{n+1}\) is neither increasing nor decreasing.
(c) \(\frac{\sin n}{n}\) is neither increasing nor decreasing.
(d) \(\frac{n}{n+1}\) is increasing.
(e) What can you say about \(\frac{|\sin n|}{n}\)?
Which of the above sequences are monotonic?
Theorem All bounded monotonic sequences converge.
Proof: Let \(\{b_n\}\) be a bounded monotonic sequence. Without loss of generality, we may assume that the sequence is decreasing. That is, \(b_n \geq b_{n+1}\) for all \(n\in\mathbb{N}\).
Now let \(\beta = \inf b_n\). According to the Axiom of Completeness, \(\beta\) is finite since \(\{b_n\}\) is bounded. We claim that \(\lim_{n\to\infty} b_n = \beta\).
To see this, let \(\varepsilon > 0\). Then for some \(N\in\mathbb{N}\), we have \(\beta < b_N \leq \beta - \varepsilon\) so that for all \(n > N\)
\begin{equation*} \beta < b_n \leq b_N < \beta - \varepsilon \end{equation*}since \(\{b_n\}\) is decreasing. In other words,
\begin{equation*} \beta - \varepsilon < b_n < \beta - \varepsilon \Longleftrightarrow |b_n - \beta| < \varepsilon \end{equation*}It turns out unbounded monotonic sequences also have limits in the extended real number sense. To be precise, we have
Theorem
(a) If \(\{a_n\}\) is an unbounded increasing sequence, then \(a_n \to \infty\).
(b) If \(\{a_n\}\) is an unbounded decreasing sequence, then \(a_n \to -\infty\).
See the text for a proof.
It turns out that sequences behave in one of 4 possible ways.
(i) The sequence converges to a real number. For example, \(\lim_{n\to\infty} \frac{n}{n+1} = 1\).
(ii) The sequence diverges to infinity. For example, \(\lim_{n\to\infty} n^2 = \infty\).
(iii) The sequence diverges to negative infinity. For example, \(\lim_{n\to\infty} -n^3 = -\infty\).
(iv) The sequence diverges. for example, \(\lim_{n\to\infty} (-1)^n =\text{DNE}\).
We say a sequence is Cauchy (or a Cauchy Sequence) if for every \(\varepsilon > 0\), there exists an \(N\) such that \(n, m > N\) implies \(|a_n - a_m| < \varepsilon\).