In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are increasing or decreasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
If a sequence of real numbers is increasing and bounded above, then its supremum is the limit.
Since is non-empty and by assumption, it is bounded above, then, by the least-upper-bound property of real numbers, exists and is finite. Now for every , there exists such that , since otherwise is an upper bound of , which contradicts to being . Then since is increasing, if . Hence, by definition, the limit of is