Monotone Sequence Convergence Theorem

Monotone sequence convergence theorem: Let $(a_n)$ be a monotone sequence of real numbers.

If $(a_n)$ is increasing and bounded above, then $(a_n)$ converges and

\lim_{n\to\infty} a_n=\sup\{a_n:\ n\in\mathbb{N}\}.

If $(a_n)$ is decreasing and bounded below, then $(a_n)$ converges and

\lim_{n\to\infty} a_n=\inf\{a_n:\ n\in\mathbb{N}\}.

Remarks

This theorem is a primary working form of the completeness axiom and is used throughout real analysis to produce limits from order and boundedness information.