Monotone and bounded sequence

A sequence that's always increasing (or always decreasing) AND stays within some fixed range — guaranteed to converge.

A sequence is monotone if it's always increasing ( $a_{n+1}\ge a_n$ for every $n$ ) or always decreasing ( $a_{n+1}\le a_n$ for every $n$ ), and bounded if it never exceeds some fixed value above (or below). The Monotone Convergence Theorem guarantees a sequence that is BOTH monotone AND bounded must converge — though the theorem doesn't identify the limit, only that one exists.

Examples

$a_n=\dfrac{n}{n+1}$ is increasing and bounded above by $1$ , so it must converge (in fact to $1$ ).