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. Informell, 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.

Inhalt 1 Convergence of a monotone sequence of real numbers 1.1 Lemma 1 1.2 Nachweisen 1.3 Lemma 2 1.4 Nachweisen 1.5 Satz 1.6 Nachweisen 2 Convergence of a monotone series 2.1 Satz 3 Beppo Levi's lemma 3.1 Satz 3.2 Nachweisen 3.2.1 Intermediate results 3.2.1.1 Lebesgue integral as measure 3.2.1.1.1 Nachweisen 3.2.1.2 "Continuity from below" 3.2.2 Beweis des Satzes 4 Siehe auch 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 {Anzeigestil (a_{n})_{nin mathbb {N} }} be such a sequence, und lass {Anzeigestil {a_{n}}} be the set of terms of {Anzeigestil (a_{n})_{nin mathbb {N} }} . Nach Annahme, {Anzeigestil {a_{n}}} is non-empty and bounded above. By the least-upper-bound property of real numbers, {textstyle c=sup _{n}{a_{n}}} exists and is finite. Jetzt, für jeden {displaystyle varepsilon >0} , es existiert {Anzeigestil N} so dass {Anzeigestil a_{N}>c-varepsilon } , since otherwise {displaystyle c-varepsilon } is an upper bound of {Anzeigestil {a_{n}}} , which contradicts the definition of {Anzeigestil c} . Dann seit {Anzeigestil (a_{n})_{nin mathbb {N} }} is increasing, und {Anzeigestil c} is its upper bound, für jeden {displaystyle n>N} , wir haben {Anzeigestil |c-a_{n}|leq |c-a_{N}|

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