Introduction

A non-decreasing sequence is not necessarily increasing if you use the standard strict definition from mathematics. The difference is whether equal neighboring terms are allowed, and that distinction matters in proofs, algorithms, and precise technical writing.

The Core Definitions

Let a_1, a_2, a_3, ... be a sequence.

The sequence is non-decreasing if every term is at least as large as the previous one. In symbols, a_{n+1} >= a_n for all valid n.

The sequence is increasing in the strict sense if every term is larger than the previous one. In symbols, a_{n+1} > a_n for all valid n.

That means repeated values are the deciding factor:

• non-decreasing allows equality
• increasing does not allow equality

So every increasing sequence is non-decreasing, but not every non-decreasing sequence is increasing.

Examples

The sequence 2, 2, 3, 3, 5 is non-decreasing because it never drops. It is not strictly increasing because it repeats values.

The sequence 1, 4, 7, 10 is strictly increasing, so it is also non-decreasing.

The constant sequence 5, 5, 5, 5 is another useful example. It is non-decreasing and non-increasing at the same time, but it is neither strictly increasing nor strictly decreasing.

These examples are simple, but they capture the whole issue.

Why Terminology Sometimes Feels Inconsistent

Some textbooks use “increasing” informally when they really mean “non-decreasing.” Others reserve “increasing” for the strict version and say “monotone increasing” or “non-decreasing” for the weak version.

That is why it is safer to prefer exact phrases such as these:

• strictly increasing
• non-decreasing
• strictly decreasing
• non-increasing

Those names remove ambiguity immediately.

Why the Difference Matters

In a proof, allowing equal terms can change what you are allowed to conclude.

For example, if a sequence is strictly increasing, then all of its terms are distinct. That statement fails for a merely non-decreasing sequence.

Consider 1, 1, 2, 3. It is non-decreasing, but it clearly does not have all distinct terms.

This distinction also appears in algorithms. If you are computing a longest increasing subsequence, you need to know whether equal values count as valid extension steps. Different definitions produce different answers.

A Small Programmatic Check

You can make the difference concrete with a short Python example.

1def is_non_decreasing(values):
2    return all(values[i] <= values[i + 1] for i in range(len(values) - 1))
3
4
5def is_strictly_increasing(values):
6    return all(values[i] < values[i + 1] for i in range(len(values) - 1))
7
8samples = [
9    [2, 2, 3, 3, 5],
10    [1, 4, 7, 10],
11    [5, 5, 5, 5],
12]
13
14for seq in samples:
15    print(seq)
16    print("non-decreasing:", is_non_decreasing(seq))
17    print("strictly increasing:", is_strictly_increasing(seq))
18    print()

Output:

1[2, 2, 3, 3, 5]
2non-decreasing: True
3strictly increasing: False
4
5[1, 4, 7, 10]
6non-decreasing: True
7strictly increasing: True
8
9[5, 5, 5, 5]
10non-decreasing: True
11strictly increasing: False

The code mirrors the mathematical definitions directly.

Related Terms

The same pattern appears on the decreasing side:

• non-increasing means a_{n+1} <= a_n
• strictly decreasing means a_{n+1} < a_n

A constant sequence is both non-decreasing and non-increasing. That often surprises people at first, but it follows immediately from the weak inequalities.

Common Pitfalls

The most common mistake is importing everyday English into formal mathematics. In casual speech, “increasing” often just means “not going down.” In technical work, many authors mean strict increase.

Another mistake is ignoring the local convention of a specific book or lecture. If an author explicitly defines “increasing” to allow equality, follow that convention in that context, but do not assume it carries over everywhere else.

A third issue is forgetting why the distinction matters. In proofs about uniqueness, injectivity, or sorted structures, strict versus weak inequalities often change the result.

Summary

• A non-decreasing sequence may repeat values
• A strictly increasing sequence may not repeat values
• Every increasing sequence is non-decreasing, but not conversely
• Constant sequences are non-decreasing without being strictly increasing
• In precise writing, say strictly increasing or non-decreasing instead of relying on ambiguous shorthand