# Monotone convergence theorem

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. Informellement, 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.

Contenu 1 Convergence of a monotone sequence of real numbers 1.1 Lemme 1 1.2 Preuve 1.3 Lemme 2 1.4 Preuve 1.5 Théorème 1.6 Preuve 2 Convergence of a monotone series 2.1 Théorème 3 Beppo Levi's lemma 3.1 Théorème 3.2 Preuve 3.2.1 Intermediate results 3.2.1.1 Lebesgue integral as measure 3.2.1.1.1 Preuve 3.2.1.2 "Continuity from below" 3.2.2 Preuve du théorème 4 Voir également 5 Notes Convergence of a monotone sequence of real numbers Lemma 1 If a sequence of real numbers is increasing and bounded above, then its supremum is the limit.

Proof Let {style d'affichage (un_{n})_{nin mathbb {N} }} be such a sequence, et laissez {style d'affichage {un_{n}}} be the set of terms of {style d'affichage (un_{n})_{nin mathbb {N} }} . Par hypothèse, {style d'affichage {un_{n}}} is non-empty and bounded above. By the least-upper-bound property of real numbers, {textstyle c=sup _{n}{un_{n}}} exists and is finite. À présent, pour chaque {displaystyle varepsilon >0} , il existe {displaystyle N} tel que {style d'affichage a_{N}>c-varepsilon } , since otherwise {displaystyle c-varepsilon } is an upper bound of {style d'affichage {un_{n}}} , which contradicts the definition of {displaystyle c} . Puis depuis {style d'affichage (un_{n})_{nin mathbb {N} }} is increasing, et {displaystyle c} is its upper bound, pour chaque {displaystyle n>N} , Nous avons {style d'affichage |c-a_{n}|leq |c-a_{N}|

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