아래로 유계인 [[감소수열,decreasing_sequence|감소수열]]과 위로 유계인 [[증가수열,increasing_sequence|증가수열]]은 수렴한다.[* http://www.kocw.net/home/cview.do?cid=ef1091e35015db7f 부산대 미적 4-4 8:27]

Every bounded, monotone sequence is convergent.[* http://www.kocw.net/home/cview.do?cid=160b162f33f362fe 최성우 12. 35m]

increasing하면서 bounded above이면 수렴.
decreasing하면서 bounded below이면 수렴. (강우석)

단조증가수열이 위로 유계이면 수렴.
단조감소수열이 아래로 유계이면 수렴.

(see [[수열,sequence#s-7]], [[유계,bounded]], [[수렴,convergence]])

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AKA '''단조수열정리'''
https://m.blog.naver.com/mindo1103/221341035050

https://everything2.com/title/monotone+sequence+theorem
https://everything2.com/title/monotone+convergence+theorem

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[[WpEn:Monotone_convergence_theorem]]
https://mathworld.wolfram.com/MonotoneConvergenceTheorem.html (short)
https://proofwiki.org/wiki/Monotone_Convergence_Theorem

[[Libre:단조_수렴_정리]]
[[WpKo:단조_수렴_정리_(수열)]]

Advanced: // 측도론 measure_theory [[측도,measure]] 까지 확장
[[WpKo:단조_수렴_정리]]
[[https://proofwiki.org/wiki/Monotone_Convergence_Theorem_(Measure_Theory)]]
https://planetmath.org/monotoneconvergencetheorem

Compare, Related: [[지배수렴정리,dominated_convergence_theorem]](writing)

Up: [[단조성,monotonicity]] [[수렴,convergence]] [[정리,theorem]]