Monotone sequence

A sequence whose terms are either nondecreasing throughout or nonincreasing throughout. Its direction may be constant, and strict monotonicity requires every successive term to change strictly in the same direction.

Sequence: An ordered list of terms indexed by natural numbers or another specified set. Monotonicity describes how successive terms in this basic object compare.
Monotone convergence theorem: A theorem stating that every bounded monotone real sequence converges. It turns the combination of monotonicity and boundedness into a limit.
Bisection method: A root-finding algorithm that repeatedly halves an interval containing a sign change. Its interval endpoints form monotone sequences converging to a root.
Strictly monotone sequence: A sequence whose terms strictly increase throughout or strictly decrease throughout. It strengthens monotonicity by forbidding equal successive terms.
Real number: An element of the continuous number system containing rational and irrational numbers. The usual monotonicity definition compares sequence terms using the real-number order.
Supremum: The least upper bound of a set in an ordered system. A bounded nondecreasing sequence converges to the supremum of its terms.
Newton's method: An iterative method that approximates a root using tangent-line updates. Under suitable conditions, monotonicity can show its iterates approach a root without overshooting.
Oscillating sequence: A sequence whose terms repeatedly move between different values or ranges rather than settling into one direction. Repeated reversals violate the defining order condition for a monotone sequence.
Inequality: A mathematical statement comparing quantities using relations such as less than or greater than. The defining condition compares each term with its successor by an inequality.
Infimum: The greatest lower bound of a set in an ordered system. A bounded nonincreasing sequence converges to the infimum of its terms.

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