Monotone sequence

A monotone sequence is a sequence $(a_n)_{n\ge 1}$ of real numbers such that either

• $a_n\le a_{n+1}$ for all $n$ (monotone increasing), or
• $a_n\ge a_{n+1}$ for all $n$ (monotone decreasing).

A sequence can be viewed as a function from $\mathbb N$ to $\mathbb R$. Monotone sequences are central in the monotone sequence convergence theorem , which combines monotonicity with being bounded above or bounded below .

Examples:

• $a_n=1-\tfrac1n$ is monotone increasing.
• $a_n=\tfrac1n$ is monotone decreasing.