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Ill SEMESTER B.Sc., , MATHEMATICS PAPER — III, , , , ANALYSIS — 1 SEQUENCES OF REAL NUMBERS, , Dr.SHIVASHARANAPPA SIGARKANTI, , ASSOCIATE PROFESSOR AND HOD, DEPARTMENT OF MATHEMATICS, GFGC K.R.PURAM, BANGALORE -560036

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GFGC K.R.PURAM III SEM B.Sc PAPER — III ANALYSIS — 1 SEQUENCES OF REAL NUMBERS, , ANALYSIS — 1 SEQUENCES OF REAL NUMBERS, , Real sequence: A real sequence is a function f:NR. and is denoted by {an}., , Ex.1. {an} = {n} = {1,2,3,... }, Ex.2. {a,} = {—n} = {-1,-2, -3,... }, Ex.3 {an} =(}=(15.5-- }, Ex.4.{a,} = ((-D}={-LL-1.. 3, , Ex.5.{a,} = {(—1)"n} = {-1,2,-3,.... }, Bounded above : A sequence {an} is said to be bounded above if there exists a, , real number M such that a, <M for alln. M is called the upper bound of {ay}, , Ex.6 ofl 12 1 ate, x. fan) = {} = {15.51 }a=2<, , {an} is bounded above., Ex.7. {ay} = {(-1)"} = {-11,-1,...},a, = (-1)" =4+1 <1, {an} is bounded above., Bounded below : A sequence {a,} is said to be bounded below if there exists a, , real number m such that a, >m foralln. m is called the lower bound of {ay}, , Ex.8.{a,} = {n} = {1,2,3, ... ja, =n2>1-. {a,} is bounded below., Ex.9 =(4.(,2 2) =i: is bounded bel, x. {an} = {=| 5 rqeqee eens} tn =e « {dn} is bounded below., , Ex.10. {a,} = {(-1)"} = {-1,1,-1,1, -1, .....},,ay = (-I)" = -1, {an} is bounded below., Bounded sequence : A sequence {ay} is said to be bounded if there exist, real numbers m andM suchthatm< a, <M forall n., i.e.if {an} is bounded above and bounded below, then {an} is called a bounded sequence, Ex.11.{a,} = {n} = {1,2,3, 0.20. 45, Qn = 1, bounded below but not above -. {a,}is not bounded, Ex.12. {ay} = {—n} = {-1,-2,-3,... }, Qn < —1,bounded above but not below -. {a,}is not bounded, , Dr.SHIVASHARANAPPA SIGARKANTI HOD OF MATHEMATICS Page 1 of 26

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GFGC K.R.PURAM III SEM B.Sc PAPER — III ANALYSIS — 1 SEQUENCES OF REAL NUMBERS, , Ex.13 .{a,} = {3 = f.5 Joo woe}, , 0 <a, <1,bounded above and bounded below -. {an}is bounded, Ex.14.{a,} = {((-1)"} = {-1,1,-1,1,-1, .....}, —1 <a, < 1,bounded above and bounded below -. {a,}is bounded, Ex.15. {a,} = {(—1)"n} = {-1,2, -3,4, 5, ....}, neither bounded below nor bounded above -..{a,} is not bounded, Least upper bound (supremum): An upper bound M is said to be the, least upper bound (l.u.b) or supremum of a sequence {an}if M is an upper bound but, M — €(for any positive number € ) is not an upper bound., Ex.16. {a,} = {—n} = {-1,-2,-3,... }, a, < —1,0,1, any positive number ,, —1,0,1, any positive number are upper bounds but l.u.bis —1, Greatest lower bound (Infimum): A lower bound m is said to be the, greatest lower bound (g.l.b) or infimum of a sequence {a,} if mis a lower bound but, m + € (for any positive number € ) is not a lower bound., Ex.17. {ay} = {n} = {1,2,3, eee, a, = 1,0,—1, any negative number, 1,0, —1, any negative number are lower bounds but g.l.b.=1, Limit of a sequence : A sequence {a,} is said to tend to limit las n tends to 0, if given e > 0 there exists a positive integer M such that |a, —I|<eWVn=>M, , Notation: lim a, =1, noo, , 1, Ex.18.Show that lim —=0, , non, i 1 1 1, Soln: |--o|<es=< €=>n>— chooseM >n n € €, , Ex.19.Show that lim 2@—+=3, Seren OW Ende ea ES., , , , Dr.SHIVASHARANAPPA SIGARKANTI HOD OF MATHEMATICS Page 2 of 26

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GFGC K.R.PURAM III SEM B.Sc PAPER — III ANALYSIS — 1 SEQUENCES OF REAL NUMBERS, , Ex.25: limn=0o . {n} is divvergent, n>, Oscillates finitely:, , A sequence {a,} is said to oscillate finitely if its limit is finite but not unique, , Ex.26: lim(—1)" =+1_ «. {(-1)"} oscillates finitely, n>, Oscillates infinitely:, A sequence {ay} is said to oscillate infinitely if its limitis +0, Ex.27: lim(—1)"n = +00 « {(—1)"n} oscillates infinitely, n>, Theorem 1. Show that every convergent sequence has a unique limit, Proof: Let {a,} be a convergent sequence. If possible let {a,} tends to two dif ferent limits, l,andl,., , lim a, =l, Given e>0 there exists positive integer M, such that, noo, , la, -h|<eforalln>M,, , lima, =l, Givene>0 there exists positive integer Mz such that, , noo, lay — l2| < e€ forall n> My)., , Let M = maximum(M,, M2), , la, -4|<eforalln>M and |a,-1l,|<eforn>M, , Consider |ly — 1,1 = |(@n — 4.) — (Qn — La) S lan — LI + lan - Lal, <e+¢é=2¢ < |l,—1,| (choose 2 < |l, —1|), i.e. |l, —1,| < |l, —l,| whichis a contradiction , hence the limit is unique, Theorem 2: Show that every convergent sequence is bounded., , Proof : Let {a,} be a convergent sequence., , lima, =1l Givene>0 there exists positive integer M such that, , n>0, la, -—I|<eforalln>M, =l-e<a,<lt+e foralln=M., , Let b = minimum{ a1, 2,03, ...@y_1,1 — €} and, , Dr.SHIVASHARANAPPA SIGARKANTI HOD OF MATHEMATICS Page 4 of 26