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# Class Note for MATH 1432 at UH 2

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Date Created: 02/06/15

Lecture 17 Section 101 Least Upper Bound Axiom Section 102 Sequences of Real Numbers Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu httpmathuhedumjiwenheMath1432 39 M e M MzsupS gt V6gtO M 6M 57 MamhllZM8 116 Math 1432 Section 26626 Lecture 17 Jiwen He University of Houston uuzmmz nztun numbes numbers mum mm mm mugs mum mm mm an mmmms a In W numbers z Quomw mm mm mum mm mm an mmmms a In W numbers z Quomw mm mm sways mum mm mm mm mm a In W numbers z Quomw mm mm gymny a x51 gymny 9 xy mum mm mm mm mm a In W numbers z Quomw mm mm gymny a x51 gymny 9 xy mum mm mm mm mm a In W numbers z Quomw mm mm gymny a x51 gymny 9 xy wwek e XSYuvny m2 zam mm xiw m mm 5 New mmmms 79 numbers z Quomw mm mm gymny a x51 nxgyzndySX 9 xzy wwek e XSYuvny axsyzndxelk e XXSY1 m2 zam mm xiw m mm 5 New mmmms 79 numbers z Quomw mm mm gymny a x51 gymny 9 xy aways e XSYuvny axsyzndxelk e XXSY1 qumyzu e was a wwww aneusmwmw m E 2nd helm hungquotth summing m m E 2nd helm hungquotth summing m Eu R m E 2nd helm hungquotth summing m Eu R En m E 2nd helm hungquotth summing m Eu R En OVXEE VyeF weMvExSy vXgtu VygtE aneumwww m E 2nd helm hungquotth summing m Eu R En anEEVyEEweMwEXSY mm azemsumm xSx Ver 27d 15y We r kmmm EN 2 m g m hm mam mmmm N am 2 ago megmmmwy mm 316mm xgx We m 19va ham Llnpzlayuz mm m nonemviysuhset s mum 2n MWEV bound mg 2 m We ham 2 ma WWW m She mnemvtysubsa 5mm M m She mnemvtysubsa Sun a s s boundm 2m iM eRsmhihzingmone M 5 a 2n MWEV bound cm s M m She mnemvtysubsa Sun a s s boundm 2m iM eRsmhihzingmone M 5 a 2n MWEV bound cm s u s sbuundm hem imam Mix quotNuner m Ema 2n wevboundmvi no W m She mnemvtysubsa Sun a s s boundm 2m iM eRsmhihzingmone M 5 a 2n MWEV bound cm s a s 5 boundm by M 3m 51km Mix 3 mmnH x e s m Ema 2n wevboundmvi a s sboundid m s bounwzbwgznd hem r y l mzswwmrs mam w 1 03 w m She mmmvtysubsa Sun a s s mama 2m NSM eRsmhthztxgvlmone M 5 Qua in ppm hm cm s a s 5 mama be m N am 6mm am 3 mmnH x e s m ma 2quot wbmms a s mm m s boummm helm rr V ngvuwwwq waneY gm nonempzysum s mum in ppm mm m 2 m We mm m ma swam Fm 1 03 w m She mmmvtysubsa Sun a s s mama 2m NSM eRsmhthztxgvlmone M 5 Qua in ppm hm cm s a s 5 mama be m N am 6mm am 3 mmnH x e s m ma 2quot wbmms a s mm m s boummm helm 55qu am mm m nonemviysuhset s mum 2n MWEV bound mg 2 m my ham 2 ma WWW Fm v Lakwgbmmgm g 1 03 w m She mmmvtysubsa Sun a s s mama 2m NSM eRsmhthztxgvlmone M 5 Qua in ppm hm cm s a s 5 mama be m N am 6mm am 3 mmnH x e s m ma 2quot wbmms a s mm m s boummm helm 55qu am mm W mmpwsubg s mum in We mm m 2 m We mm 2 ma Wm Fm m kappa 50mm 32nd g Newt no w m She mnemvtysubsa Sun a s s boundm 2m iM eRsmhihzingmone M 5 a 2n MWEV bound cm s a s 5 boundm by M 3m 51km Mix 3 mmnH x e s m Ema 2n wevboundmvi a s sboundid m s bounwzbwgznd hem 55qu am mm m nonemviysuhset s mum 2n MWEV bound mg 2 m my ham 2 ma WWW Fm m kappa 50mm 32nd g Newt WEE g e mm a ibeksuzhihzixsb Vxefzndbsy 1 03 w m She mmmvtysubsa Sun a s s boundm 2m iM eRsmhihzingmone M 5 a 2n MWEV bound cm s a s 5 boundm by M 3m 51km Mix 3 mmnH x e s m Ema 2n wevboundmvi a s sboundid m s bounwzbwgznd hem Ramaw am musw w nonemviysuhset s mum 2n MWEV bound mg 2 m my ham 2 ma WWW Latfz ppabourdsfuv zndf g e um Newt e ibeksuzhihzixsb Vxefzndbsy VY Kb52b2nuvvmbuunddi 1 03 w m She mmmvtysubsa Sun a s s boundm 2m iM eRsmhihzingmone M 5 a 2n MWEV bound cm s a s 5 boundm by M 3m 51km Mix 3 mmnH x e s m Ema 2n wevboundmvi a s sboundid m s bounwzbwgznd hem Ramaw am musw w nonemviysuhset s mum 2n MWEV bound mg 2 m my ham 2 ma WWW Latfz ppabourdsfuv zndf g e um Newt e ibeksuzhihzixsb Vxefzndbsy WEF b szbznuppevboundoS e mmgmms Real Numbers Sequences Supremum or Infimum of a Set 5 Definition LJ Definition r 39 771 Jiwen He University of Houston Real Numbers Sequences Supremum or Infimum of a Set 5 Definition Let S be a nonempty subset of R with an upper bound We denote by supS or ubS the supremum or least upper bound of S LJ Definition r 39 771 Jiwen He University of Houston Real Numbers Sequences Supremum or Infimum of a Set 5 Definition Let S be a nonempty subset of R with an upper bound We denote by supS or ubS the supremum or least upper bound of S LJ Definition r 39 771 Jiwen He University of Houston Real Numbers Sequences Supremum or Infimum of a Set 5 Definition Let S be a nonempty subset of R with an upper bound We denote by supS or ubS the supremum or least upper bound of S Let M supS Then LJ Definition r 39 771 Jiwen He University of Houston Real Numbers Sequences Supremum or Infimum of a Set 5 Definition Let S be a nonempty subset of R with an upper bound We denote by supS or ubS the supremum or least upper bound of S Let M supS Then 0 X g M VX E 5 LJ Definition r 39 771 Jiwen He University of Houston Real Numbers Sequences Supremum or Infimum of a Set 5 Definition Let S be a nonempty subset of R with an upper bound We denote by supS or ubS the supremum or least upper bound of S Let M supS Then 0 X g M VX E S oVegtO M eM 57Z LJ Definition r 39 771 Jiwen He University of Houston Supremum or Infimm of a Set 5 n Let S be a nonempty subset of R with an upper bound We denote by 1 or39 V the or m of S Let M supS Then f 7 j V J h 3 M8 M X g M VX E S CVegt0 M 6M 57 eff n n Let S be a nonempty subset of R with a lower bound We denote by or the or 7 015 IT Supremum or Infimm of a Set 5 n Let S be a nonempty subset of R with an upper bound We denote by 1 or39 V the or m of S j Let M suPS39 Then X S M VX E S C Vegt07 m 67M s LJ eff n n Let S be a nonempty subset of R with a lower bound We denote by or the or 7 015 quot L nfn Let m infS Then IT Supremum or Infimm of a Set 5 n Let S be a nonempty subset of R with an upper bound We denote by 1 or39 V the or m of S Let M supS Then x g M VX E S CVegt0 M 6M 57 f 7 j V J h LJ eff n n Let S be a nonempty subset of R with a lower bound We denote by or the or 7 015 ann Let m inf5 The cquot X 2 my VX E 5 IT Supremum or Infimm of a Set 5 n Let S be a nonempty subset of R with an upper bound We denote by 1 or39 V the or m of S j Let M suPS39 Then X S M VX E S C Vegt07 m 67M s LJ eff n n Let S be a nonempty subset of R with a lower bound We denote by or the or 7 015 ann Let m inf5 The i sz it V gt07 m7m s IT mum n a m We sum mm m both wanna My how SW1 Mum I M m4 much 0 a m We sum mm m both wanna My how may 3 wall 1 w a lt b 09quot b min mph 2nd a z mm mun I M m4 much 0 a m We sum mm m both wanna My how supn z zpz mmua Wzltthen b suvz b5upz b dazm z bl uW 70260 qumenws my suv s 7 I M m4 much 0 a m We sum mm m both wanna My how may 3 wall 1 w a lt b 09quot 7 WM mph m z MM my aw 4qu gamems supszw w x6 8lt7r 0enm57 asupsz I M m4 much 0 a m We sum mm m both wanna My how may 3 wall 1 w a lt b 09quot 7 WM mph m z MM my aw 4qu gamems supszw w xelk3lt1r thenm5 7 my aw er8ltwr zhenm15 73 5uv5 I M m4 much 0 a m We sum mm m both wanna My how may 3 wall 1 w a lt b 09quot 7 WM mph m z MM my aw 4qu gamems supszw w xelk3lt1r thenm5 7 my aw er8ltwr zhenm15 73 5uv5 m Mm WWWquot Wampum an mm m MUM that MS 7 7mm 4 6 I when 7 A Ememw quotmy 5 2 quotEmma Wm gum on the 53 m pomE M235 1 w mm 3 in z 390 all We m m m m 5 dam by n Maw A Ememw quotmy 5 2 quotEmma Wm gum on the 53 m pomE M235 1 w mm 3 in z 390 all We m m m m 5 dam by n The m nzen 2 sdem ad by whim in A Ememw quotmy 5 2 quotEmma Wm gum on the 53 m pomE M235 1 w mm 3 in z 390 all We m m m m 5 dam by n The m nzen 2 sdem ad by whim in quotENquot sihe Enema L g g 1 I M m4 Maw 1 A Ememw quotmy 5 2 quotEmma Wm gum on the 53 m pomE M235 1 seal pm We m m m m 5 dam by n 9 en The m nzen 2 sdem ad by whim in I M m4 Maw 1 A Ememw quotmy 5 2 quotEmma Wm gum on the 53 m pomE M235 1 seal pm We m m m m 5 dam by n 9 en The m nzen 2 sdem ad by whim in quote a We mm L g g n z new meme I M m4 Maw 1 A Ememw quotmy 5 2 quotEmma Wm gum on the 53 m pomE M235 1 seal pm We m m m m 5 dam by n 9 en The m nzen 2 sdem ad by whim in quotENquot sihe quemet 1 M z new mam e a newerquotsihemmnzehkg t 4m quot6quot sihesiqueme 4 L minm bezmmommmbes AreahmmbevL sz WWWL demi d by V5gtEL aweummqumww minm bezmmommmbes AreahmmbevL sz WWWL demi d by V5gtEL aweummqumww minm bezmmommmbes AreahmmbevL sz WWWL demi d by V5gtEL aweummqumww new ir enhm whzu Maw M minm bezmmommmbes AreahmmbevL sz WWWL demi d by L W mu aNeusmwwqmvquotgtN uh nmmmwn runnyougm zhu NgtDsuzhihzisNgt1na 9 Then ngtN weh2EEIltI IltN Zlt N uh mama Nahum M minm bezmmommmbes AreahmmbevL sz WWWL demi d by L W mu aNeusmwwqmvquotgtN uh nmmmvh runnyougm zhu NgtDsuzhihzisNgt1na 9 Then ngtN weh2EEIltI IltN Zlt wan 1 quot6quot 09quot th minm bezmmommmbes AreahmmbevL sz WWWL demi d by L W mu aNeusmwwqmvquotgtN um ne whenhmnvzn rum gtugmambasummmm 7amp5ThenmhugtMweh2vEDlt iltw lts my ne whenhm wan rum gtugmambasummmm kgnemwmmmuq 1 w m mam m WWW L2nd hman a Assamezm M52 hm i sg dw bemmvg t mam gm mmwhzmdmw mm u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim w mm an u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim WWW M I M V5gtu 3NWWMMM hmqu 2nd hman M Men u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim mm M I Wer v gtu 3NgtugnzmzaniLkgzwaniwkg vww e LMganLanMlt 2nd hman M Men u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim mm M I Wer v gtu 3NgtugnzmzaniLkgzwaniwkg vww e LMganLanMlts e L M 2nd hman M Men u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim w mm an Wer 2nd hman M Men V5gtu 3NWWMMM hmqu e LMganLanMlts e L M an n e m we mm M 5 mm u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim w mm an Wer 2nd hman M Men V5gtu 3NWWMMM hmqu e LMganLanMlts e L M man W 4mnewmmmmhg mm M gmmemm xig g has m gm ML u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim w mm an Wer 2nd hman M Men V5gtu 3NWWMMM hmqu e LMganLanMlts e L M man W 4mnewmmmmhg mm M gmmemm xig g has m gm 5ltLVxeR mquotnmzummmdihmumm u M a Assamezm M52 hm i sg dw bemmvg t a A WNW has quota W 5 w m bede whim w mm an Wer 2nd hman M Men V5gtu 3NWWMMM hmqu e LMganLanMlts e L M man W 4mnewmmmmhg mm M gmmemm xig g has m gm M L mm mquot mth m 4 Mum mm m s quotm mm m m M m m m x g n 5 mm mm m mm bdw m wt 5 mm mm m mm 5mm ihesa 5 mm m mung 5 mm mm m mm bdw m wt 5 mm mm m mm 5mm ihesa 5 mm m mung a w iv 7 ENquot mequot Name an 5 mm mm byM 31m mm bekw m min 5 mm mm m mm bdw m wt 5 mm mm m mm 5mm ihesa 5 mm m mung a w iv 7 ENquot mequot Name an 5 mm mm byM 31m mm bekw m min m n e N m M 215 m we boundmmemmew m mg me mm m the Emma an 5 mm mm m mm bdw m wt 5 mm mm m mm 5mm ihesa 5 mm m mung a w iv 7 ENquot mequot Name an 5 mm mm byM 31m mm bekw m min m n e N m M 215 m we boundmmemmew m mg me mm m the Emma an Wang my WWW Wm 5 mm s banned mm m mm bdw m wt 5 mm mm m mm A Wm we 5mm ihesa 5 mm m Marja a w an i 7 erquot new Mame an 5 mm mm byM 3 m mama him by mSEI m n e m new M 215 in may hm cummme M m m g 4 s m m mm m the mm A 3 We 1 hmnwuanL I s banned mm m mm bdw m wt 5 mm mm m mm A Wm we 5mm ihesa 5 mm m Marja a w an i 7 erquot new Mame an 5 mm mm byM 3 m mama him by mSEI m n e m new M 215 in may hm cummme M m m g 4 s m m mm m the mm A 3 We 1 wwww V5gtEL3Nest zan lts VngtN I s banned mm m mm haw m wt 5 mm mm m mm A Wm we 5mm ihesa 5 mm m Marja a w an i 7 erquot new Mame an 5 mm mm byM 3 m mama him by mSEI m n e m new M 215 in may hm cummme M m m g 4 s m m mm m the mm A WWW 1 WW Mam w wm eMShArLHMlt HLLVngtN I s banned mm m mm haw m wt 5 mm mm m mm A Wm we 5mm ihesa 5 mm m Marja a w an i 7 erquot new Mame an 5 mm mm byM 3 m mama him by mSEI m n e m new M 215 in may hm cummme M m m g 4 s m m mm m the mm A WWW 1 WW Mam w wm a MS HM WWW w I Ufa v m A Wm am mama Mm s hm m mm s hm dzbmuvboundzd We am 5 mm mm m mm u w an 7 7 erquot new Mame an 5 mama 2m byM 3 m mama bebN w mSEI 1 neN then M2 We hm WWW n m m S 7 s in W mm a m Ewe M quot vmm W MW 3 My raw W mmmm 3 WW I a A 53mm mg 5 mmsmg w an 5 am mu n e w a A 53mm quot2 5 mmsmg w in g M mu n e w A 53mm mg 5 42mm n in 3 W cm 2 quote w nal 6 A 53mm quot2 5 mmsmg w in g M mu n e w a A 53mm mg 5 4mm n in 3 W cm 2 quote w M39Vman a A bounM mousmgxequena onvuge to 5 Ma nal 6 A 53mm quot2 5 mmsmg w in g M mu n e w a A 53mm mg 5 4mm n in 3 W cm 2 quote w 1mm a A bmw mumqu tonvuge w 5 m u a bmw Mummqm emerge w my nal 6 A 53mm quot2 5 mmsmg w in g M mu n e w a A 53mm mg 5 dxrezsmg n in 3 W cm 2 quote w M39Vman a A bounM mousmgxequena onvuge to 5 Ma l u a bmw Mummqm emerge w my quot6quot 09quot an Eda925mg boun d 271 man u anquot i nal 6 A 53mm quot2 5 mmsmg w in g M mu n e w a A 53mm mg 5 dxrezsmg n in 3 W cm 2 quote w rmaa a A bounM mousmgxequena onvuge to 5 Ma l u a bmw Mummqm emerge w my new men 3quot 549mm bounfad m man u 2 f lt1 anquot i nal 6 A 53mm quot2 5 mmsmg w in g M mu n e w a A 53mm mg 5 dxrezsmg n in 3 W cm 2 quote w rmaa a A bounM mousmgxequena onvuge to 5 Ma l u a bmw Mummqm emerge w my quot6quot 09quot an mam bounfad 27d mm a 2 a win quota new new A 5 mm m Mama immimmmm m Swan m Sequences 1 7 7 7 7 7 7 77777777i7 07T7 O l o 2 01 a2 a3 16 I J J JJ I I I I I I I O 1 a 31 1 1 2 3 4 5 6 7 n 2 3 4 5 Jiwen He University of Houston M th 14432 quotS miqw Leeture 17 Sequences 1 7 7 7 7 7 7 77777777i7 07T7 O O l o 2 51 12 13 16 I J J JJ I I I I I I I O 1 a 31 1 1 2 3 4 5 6 7 n 2 3 4 5 Example n Let an nl HEN Jiwen He University of Houston M th 14432 ec an26626 Leeture 17 Sequences 1 7 7 7 7 7 7 77777777i7 07T7 O O l o 2 51 12 13 16 I J J JJ I I I I I I I O 1 a 31 1 1 2 3 4 5 6 7 n 2 3 4 5 Example n gtilt Let an nl HEN 0 an is increasing Jiwen He University of Houston M th 14432 Sec qni e Leeture 17 MarChI 11 12 Sequences 1 7 7 7 7 7 7 77777777i7 07T7 O l o 2 01 a2 a3 616 J J JJ I I I I I I I 1 a 31 1 1 2 3 4 5 6 7 n 2 3 4 5 Let an an is increasing lt2 nEN an1 3n n2 n Jiwen He University of Houston March 11 2608 Math 1m Sectiqnime Lecture 17 1216 Sequences Q Q N D 6 ll mml IIt D DJN D n gtk Let an n1n N an1 n1 n1 n22nl 0 an IS Increasmg lt2 an n n 22 gt 1 2 1 2 3 99 o The sequence displays as 5 g Z Jiwen He University of Houston M th 14432 Sec qn Leeture 17 Mal39Ch 11 Sequences Q Q N D 6 IL I N wlm Let an nl EN 2 0 an IS Increasmg lt2 m1 n1 n 2nl gt 1 an n2 7 22 0 The sequence displays as g gt supan 1 and infan Jiwen He University of Houston Math 1432 Sectiqnquot26623967 Laeture 17 March 11quot Sequences Q Q N D 6 IL I m LuN n gtk Let an n1n N 2 0 an IS Increasmg lt2 32 1 inl gt 1 1 2 3 99 o The sequence displays as 5 g Z m gt supan 1 and infan gt Iim an supan 1 H n gtoo Jiwen He University of Houston M th 14432 Sec qn e Leeture 17 MarCh 11 Example Sequences an 2 00 0 1 2 a43 I9 123456 n Jiwen He University of Houston Math 1432 Section 26626 Lecture 3917 March 11 2008 13 16 Sequences Example 2 0 o Letan2 withnnn 11 O 1 614 3 III I 9 123456 n Jiwen He University of Houston Math 1432 Section 26626 Lecture 3917 March 11 2008 13 16 Sequences Example 2 0 o Letan2 with nnn 11 0 an is decreasing 1 614 3 I I I I i 9 12 3 4 5 6 n Jiwen He University of Houston Math 1432 Section 26626 Lecture 3917 March 11 2008 13 16 Sequences Example 2 00 Letan2 withnnn 11 0 an is decreasing 39 a 2n1 n 2 1 lt quot1 lt 1 an n1 n1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 3917 March 11 2008 13 16 Sequences Example 2 00 Letan2 withnnn 11 0 an is decreasing 2 1 lt2 an n lt 1 an n i l2quot n1 a42 0 supan 2 and infan O I I I I i 9 1 2 3 4 5 6 n Jiwen He University of Houston Math 1432 Section 26626 Lecture 3917 March 11 2008 13 16 Sequences Example 2 00 Letan2 withnnn 11 0 an is decreasing a 2 1 4 quot1 n lt 1 an n i 12n n i 1 a42 0 supan 2 and infan O I I I I i 9 1 2 3 4 5 6 n gt Iim an infa O n gtoo Jiwen He University of Houston Math 1432 Section 26626 Lecture 3917 March 11 2008 13 16 Sequences Example Lal 39 6x 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 14 16 Sequences Example Let an Lal 39 6x 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 14 16 Sequences Example Let an 0 an is decreasing Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 1 1416 Sequences Example Let an 0 an is decreasing lt2 Let fX Lal 39 6x I x F Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 14 16 Sequences Example Let an 0 an is decreasing y lt2 Let fX V 1611 yz 2 a2 X X sag Mia e X6 1 X I I I 4 f X Z 2 lt O 1 2 3 4 5 X e X eX I x F Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 14 16 Sequences Example Let an 0 an is decreasing y lt2 Let fX V 1611 yz 2612 X X 3a3 Mia e X6 1 X I I I 4 f X I f lt O 1 2 3 4 5 X e X eX o supan i and infan O I x F Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 14 16 Sequences Example Let an 0 an is decreasing lt2 Let fX eX XeX 1 X 4a I I I 4 fX lt O o supan i and infan O gt Iim an infan O n gtoo I Jiwen He University of Houston Math 1432 Section 26626 Lecture 17 March 11 2008 14 16 my nzm a an sdaoezsmgfuv quotz my nzm a an sdaoezsmgfuv quotz m m gnaw a an sdaoezsmgfuv quotz m X ux gnaw m z A WMmx m lt u a an sdaoezsmgfuv quotz m X ux gnaw m z A WMmx m lt u a man 1 a an sdaoezsmgfuv quotz m X ux gnaw m z A WMmx m lt u a man 1 e Aka zmqa a Pa Numbers Pen2w 4 Last um Baum a S uemsm s mm My

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