Monotone Convergence Theorem
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Monotone convergence theorem From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the monotone convergence theorem is any of several theorems. Some major examples are presented here. Contents [hide] • 1 Convergence of a monotone sequence of real numbe rs o 1.1 The orem o 1.2 Pr oof o 1.3 Remar k • 2 Convergence of a monotone seri es o 2.1 The orem • 3 Lebesgue 's monotone conver gence theorem o 3.1 The orem o 3.2 Proof • 4 See als o • 5 Note s [edit] Convergence of a monotone sequence of real numbers [edit] Theorem If a n is a monotone sequence of real numbers (e.g., if a n ≤ a n+1 ), then this sequence has a finite limit if and only if the sequence is bound ed. [1] [edit] Proof We prove that if an increasing sequence is bounded above, then it is convergent and the limit is . Since is non-empty and by assumption, it is bounded above, therefore, 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,
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