A monotonic sequence is a sequence of numbers that moves in only one direction: either always going up or always going down. More precisely, a sequence is monotonic if each term is greater than or equal to the one before it (non-decreasing), or each term is less than or equal to the one before it (non-increasing). This simple property turns out to have powerful consequences for whether a sequence settles on a final value.
The Core Idea
Think of a monotonic sequence as a one-way street. The sequence 1, 2, 3, 4, 5, … only goes up. The sequence 10, 7, 4, 1, … only goes down. Neither one ever reverses direction, and that’s what makes them monotonic.
A sequence that bounces back and forth, like 1, 3, 2, 4, 3, 5, is not monotonic because it doesn’t commit to a single direction. The classic alternating sequence 1, −1, 1, −1, … also fails the test for the same reason.
Four Types of Monotonic Behavior
Monotonic sequences come in four flavors, depending on how strict the direction is:
- Strictly increasing: each term is larger than the previous one (no ties allowed). Example: 2, 5, 11, 20, …
- Non-decreasing: each term is greater than or equal to the previous one (ties are fine). Example: 1, 1, 2, 2, 3, 3, …
- Strictly decreasing: each term is smaller than the previous one. Example: 100, 50, 25, 12.5, …
- Non-increasing: each term is less than or equal to the previous one. Example: 5, 5, 4, 4, 3, 3, …
Any sequence fitting one of these four categories counts as monotonic. The “strictly” versions are a subset of the more relaxed versions. A strictly increasing sequence is automatically non-decreasing, but a non-decreasing sequence isn’t necessarily strictly increasing because it might contain repeated values.
How to Test Whether a Sequence Is Monotonic
If your sequence is defined by a formula, there are two practical approaches. The first is to compare consecutive terms directly. Compute the difference between term n+1 and term n. If that difference is always positive, the sequence is increasing. If it’s always negative, the sequence is decreasing.
The second method works when the sequence comes from a continuous function. If you can write the terms as f(n) for some function f, take the derivative of f. A positive derivative over the relevant range means the function is increasing, which tells you the sequence is increasing too. A negative derivative means it’s decreasing. This derivative approach is a standard technique taught in calculus courses and often appears on homework for exactly this purpose.
For sequences defined by a recurrence relation (where each term depends on the previous one), you may need to use induction: show the pattern holds for the first term, then prove that if it holds for term n, it must also hold for term n+1.
Why Monotonicity Matters: The Convergence Theorem
The most important result about monotonic sequences is the Monotone Convergence Theorem. It says: if a sequence is monotonic and bounded, it converges. That’s it. Two conditions, one guarantee.
“Bounded” means the sequence doesn’t grow without limit. An increasing sequence is bounded if there’s some ceiling it never exceeds. A decreasing sequence is bounded if there’s some floor it never drops below. When both conditions are met, the sequence must settle toward a specific finite number.
The theorem also tells you exactly what the sequence converges to. A bounded increasing sequence converges to its least upper bound, the smallest number that sits above every term in the sequence. A bounded decreasing sequence converges to its greatest lower bound, the largest number that sits below every term. For example, the sequence 1/2, 3/4, 7/8, 15/16, … is increasing and never exceeds 1. The Monotone Convergence Theorem guarantees it converges, and its limit is 1.
This theorem is one of the cornerstones of calculus and mathematical analysis. It gives you a way to prove a sequence has a limit without needing to find the limit first. You just need to show two things: the terms keep moving in one direction, and they don’t blow up to infinity.
What Happens Without a Bound
If a monotonic sequence is unbounded, the behavior is equally predictable, just in the opposite way. An increasing sequence with no upper bound diverges to positive infinity. A decreasing sequence with no lower bound diverges to negative infinity. There’s no middle ground for monotonic sequences: they either converge to a finite limit or head off to infinity. This clean either-or behavior is part of what makes them so useful in proofs.
The reasoning is straightforward. Take an increasing sequence with no upper bound. Pick any number, no matter how large. Because the sequence is unbounded, some term eventually exceeds that number. And because the sequence is increasing, every term after that one is even larger. So the sequence surpasses any target you set, which is exactly what diverging to infinity means.
Monotonic Subsequences
Even sequences that aren’t monotonic contain monotonic pieces hiding inside them. A famous result in combinatorics (sometimes called the Erdős–Szekeres theorem) guarantees that any sequence of more than n² distinct numbers contains either an increasing subsequence or a decreasing subsequence of length greater than n. This means long sequences inevitably harbor significant monotonic structure, regardless of how chaotic they look on the surface.
Finding the longest increasing or decreasing subsequence within a larger sequence is a classic problem in computer science. Efficient algorithms exist to identify these subsequences, and they show up in applications from sorting to data analysis. One well-known approach, Schensted’s algorithm, processes elements one at a time and maintains a running record that implicitly tracks the longest increasing subsequence seen so far.
Common Examples Worth Knowing
The sequence 1/n (that is, 1, 1/2, 1/3, 1/4, …) is strictly decreasing and bounded below by zero. By the Monotone Convergence Theorem, it converges, and its limit is 0.
The sequence defined by repeated square roots, where each term equals the square root of 2 plus the previous term, is increasing and bounded above by 2. It converges to 2. Sequences like this frequently appear in textbook exercises because proving they converge requires exactly the monotone-and-bounded strategy.
Partial sums of a series with positive terms form an increasing sequence. This is why the Monotone Convergence Theorem shows up constantly in the study of infinite series: determining whether a series converges often reduces to checking whether its partial sums stay bounded.