Monotone sequence

A real sequence $(a_n)_{n\in\mathbb{N}}$ is monotone if it is either:

• nondecreasing: $a_{n+1}\ge a_n$ for all $n$, or
• nonincreasing: $a_{n+1}\le a_n$ for all $n$.

Monotone sequences are central in real analysis because bounded monotone sequences converge in $\mathbb{R}$ (monotone convergence theorem for sequences).

Examples:

• $a_n = 1 - \frac{1}{n}$ is nondecreasing and converges to $1$.
• $a_n = \frac{1}{n}$ is nonincreasing and converges to $0$.
• $a_n = (-1)^n$ is not monotone.