A sequence can be hard to understand because it gives infinitely many pieces of information one term at a time. Even if the first thousand terms look stable, the next term could jump away unless some extra structure prevents it. Monotonicity is the simplest such structure: it says the sequence is allowed to move only in one direction. Once a sequence has chosen its direction, the only remaining question is whether something stops it.

The guiding picture is an approximation process. Suppose a decimal expansion is revealed by truncating more and more digits. The approximations from below never decrease, and they are trapped above by the number being approximated. The convergence does not come from a closed formula for the terms; it comes from order together with a bound. This is the prototype behind many existence arguments in analysis.

[example: Decimal Truncations from Below] Let $x = 1.4142135\ldots$, and define \begin{align*} a_n = \frac{\lfloor 10^n x \rfloor}{10^n}. \end{align*} Write $m_n=\lfloor 10^n x\rfloor$. By the defining property of the floor function, \begin{align*} m_n \le 10^n x < m_n+1. \end{align*} Multiplying by $10$ gives \begin{align*} 10m_n \le 10^{n+1}x < 10m_n+10. \end{align*} Since $\lfloor 10^{n+1}x\rfloor$ is the greatest integer less than or equal to $10^{n+1}x$, the inequality $10m_n \le 10^{n+1}x$ implies \begin{align*} 10m_n \le \lfloor 10^{n+1}x\rfloor. \end{align*} Dividing by $10^{n+1}$, we get \begin{align*} a_n = \frac{m_n}{10^n} = \frac{10m_n}{10^{n+1}} \le \frac{\lfloor 10^{n+1}x\rfloor}{10^{n+1}} = a_{n+1}. \end{align*} Thus $(a_n)$ is increasing. The same floor inequality also gives the upper bound and the error estimate. Dividing \begin{align*} m_n \le 10^n x < m_n+1 \end{align*} by $10^n$ gives \begin{align*} a_n \le x < a_n+10^{-n}. \end{align*} Subtracting $a_n$ throughout yields \begin{align*} 0 \le x-a_n < 10^{-n}. \end{align*} Given $\varepsilon>0$, choose $N\in\mathbb N$ such that $10^{-N}<\varepsilon$. If $n\ge N$, then $10^{-n}\le 10^{-N}$, so \begin{align*} 0 \le x-a_n < \varepsilon. \end{align*} Hence $a_n\to x$. The truncations approach $x$ from below: monotonicity supplies the one-way motion, and the estimate $x-a_n<10^{-n}$ shows that the remaining gap vanishes. [/example]

The lesson is that monotone sequences turn convergence into an order question. Instead of estimating all pairs $|a_m-a_n|$, we ask whether the sequence has a ceiling or a floor. This makes monotone sequences a bridge between the definition of a [sequence](/page/Sequence), the order structure of $\mathbb R$, and the completeness properties developed in [Cambridge IB Analysis and Topology](/page/Cambridge%20IB%20Analysis%20and%20Topology).

## Definition

### Monotone Directions

A general sequence may move up, down, and back again. The page topic is the special case where this never happens: after any term, the next term is constrained to lie on the same side in the order on $\mathbb R$. That single restriction is strong enough to turn many convergence questions into questions about upper and lower bounds.

[definition: Monotone Sequence] Let $(a_n)_{n=1}^{\infty}$ be a sequence of [real numbers](/page/Real%20Numbers). The sequence is monotone if either \begin{align*} a_n \le a_{n+1} \end{align*} for every $n \in \mathbb N$, or \begin{align*} a_n \ge a_{n+1} \end{align*} for every $n \in \mathbb N$. [/definition]

This definition is a child of the general notion of a sequence: monotonicity adds an order condition to the existing data of indexed real terms. It does not require a formula for $a_n$, and it does not assert convergence by itself. To talk efficiently about the two possible directions, we now name them separately.

The upward direction is the order-theoretic version of approaching a target from below. It is the form most directly connected to suprema, because the terms climb toward the least ceiling that contains all of them.

[definition: Increasing Sequence] Let $(a_n)_{n=1}^{\infty}$ be a sequence of real numbers. The sequence is increasing if \begin{align*} a_n \le a_{n+1} \end{align*} for every $n \in \mathbb N$. [/definition]

Some approximation schemes begin above the desired quantity and improve by moving downward. To treat upper barriers, overestimates, and decreasing interval endpoints with the same precision, we need the reverse order condition as its own named notion.

[definition: Decreasing Sequence] Let $(a_n)_{n=1}^{\infty}$ be a sequence of real numbers. The sequence is decreasing if \begin{align*} a_n \ge a_{n+1} \end{align*} for every $n \in \mathbb N$. [/definition]

With this terminology, a monotone sequence is exactly a sequence that is increasing or decreasing. Many results have one proof for increasing sequences and a second proof obtained by changing signs for decreasing sequences.

### Strict Monotonicity

Sometimes a sequence moves in one direction and never repeats a value. We need a stricter word for situations where the terms themselves must be distinct, such as constructing infinite ordered subsets or proving that a sequence never stabilizes.

[definition: Strictly Increasing Sequence] Let $(a_n)_{n=1}^{\infty}$ be a sequence of real numbers. The sequence is strictly increasing if \begin{align*} a_n < a_{n+1} \end{align*} for every $n \in \mathbb N$. [/definition]

Strict increase is not the only strict order pattern that occurs in approximation. An algorithm may produce genuinely smaller overestimates at every stage, and in such a case plateaus are ruled out from above rather than below. Naming the downward strict version keeps later examples from blurring strict decrease with ordinary decrease.

[definition: Strictly Decreasing Sequence] Let $(a_n)_{n=1}^{\infty}$ be a sequence of real numbers. The sequence is strictly decreasing if \begin{align*} a_n > a_{n+1} \end{align*} for every $n \in \mathbb N$. [/definition]

A strictly increasing sequence is increasing, and a strictly decreasing sequence is decreasing. The converse fails because constant stretches are allowed in monotone sequences.