monotone sequence

A monotone sequence only ever moves one way. Picture a hiker who either never steps downhill or never steps uphill: a monotone sequence either never decreases or never increases as the index advances. It may pause on a plateau (repeat a value) but it never reverses direction.

Formally, (a_n) is increasing if a_n <= a_{n+1} for all n, and decreasing if a_n >= a_{n+1} for all n; in either case it is called monotone. When the inequalities are strict (a_n < a_{n+1}, or a_n > a_{n+1}), the sequence is strictly increasing or strictly decreasing. The word 'non-decreasing' is often used as a synonym for increasing to stress that equal consecutive terms are allowed.

Monotonicity is prized because it tames the wildness of limits. For a general sequence you must rule out oscillation to prove convergence, but a monotone sequence cannot oscillate — it can only climb steadily, fall steadily, or level off. Combined with boundedness, this forces convergence outright, which is the content of the monotone convergence theorem.