Monotone convergence theorem

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 non-decreasing or non-increasing) 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. Let be such a sequence, and let be the set of terms of . By assumption, is non-empty and bounded above. 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 the definition of . Then since is increasing, and is its upper bound, for every , we have . Hence, by definition, the limit of is If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit. The proof is similar to the proof for the case when the sequence is increasing and bounded above. If is a monotone sequence of real numbers (i.e., if an ≤ an+1 for every n ≥ 1 or an ≥ an+1 for every n ≥ 1), then this sequence has a finite limit if and only if the sequence is bounded. "If"-direction: The proof follows directly from the lemmas. "Only If"-direction: By (ε, δ)-definition of limit, every sequence with a finite limit is necessarily bounded. If for all natural numbers j and k, aj,k is a non-negative real number and aj,k ≤ aj+1,k, then The theorem states that if you have an infinite matrix of non-negative real numbers such that the columns are weakly increasing and bounded, and for each row, the series whose terms are given by this row has a convergent sum, then the limit of the sums of the rows is equal to the sum of the series whose term k is given by the limit of column k (which is also its supremum).