Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A monotone sequence is a sequence of numbers that is either entirely non-increasing or non-decreasing. In other words, the terms either consistently increase or consistently decrease.
5 Must Know Facts For Your Next Test
A sequence $\{a_n\}$ is monotone increasing if $a_{n+1} \geq a_n$ for all $n$.
A sequence $\{a_n\}$ is monotone decreasing if $a_{n+1} \leq a_n$ for all $n$.
Monotone sequences are always bounded by their initial term and any subsequent terms.
The Monotone Convergence Theorem states that every bounded monotone sequence converges.
To prove a sequence is monotone, you can show that the difference between consecutive terms always has the same sign.
Review Questions
Related terms
Bounded Sequence: A sequence $\{a_n\}$ is bounded if there exists a real number $M$ such that $|a_n| \leq M$ for all $n$.
Convergent Sequence: A sequence $\{a_n\}$ converges to a limit $L$ if for every $\epsilon > 0$, there exists an integer $N$ such that for all $n \geq N$, $|a_n - L| < \epsilon$.
Subsequence: A subsequence is derived from another sequence by deleting some or none of its elements without changing the order of the remaining elements.