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Explanation:- counter example: in Lan) = = (-1) :nGN sequencence of real number which is not convergent it is oscillatory. oscillate finitely be) Proofin let two sequence g so given 430 I, and I be the points of Land and [bn] respectively, J no G12 sit we have 2-₁₁ -1₁ -1, 1,-1,---- y 2-1,011 2 Nyno cans - dil 22 and | 2bn3 - 12/2€. 2 Lan) Hơn 3) kith 8 our claim 1.8. [an] tibn) cyf to i+dz ✓ N», no. we will use triangle inequality V equality. stan an-dil 올 가을 2 Bum of two 2 & ... kandbon) I LE Hence Convergent sequence is convergent. + 1 barda]

Explanation:- counter example: in Lan) = = (-1) :nGN sequencence of real number which is not convergent it is oscillatory. oscillate finitely be) Proofin let two sequence g so given 430 I, and I be the points of Land and [bn] respectively, J no G12 sit we have 2-₁₁ -1₁ -1, 1,-1,---- y 2-1,011 2 Nyno cans - dil 22 and | 2bn3 - 12/2€. 2 Lan) Hơn 3) kith 8 our claim 1.8. [an] tibn) cyf to i+dz ✓ N», no. we will use triangle inequality V equality. stan an-dil 올 가을 2 Bum of two 2 & ... kandbon) I LE Hence Convergent sequence is convergent. + 1 barda]

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 78E
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Explanation: counter example in Lan) = (-1) 9.) = 2-1, 41,-1,1,-1,-- y :nGN sequencence of real number which is not it is oscillatory. convergent it is oscillate finitely bu) Proof: - let two sequence g 30, I, and Is V given Exo Nyno be the points of Land and Lbn> respectively. I no G Z sif 1 our claim rs. we have Ny ho. Lan) - do I Le and 2 [an] tibn) cyf to dit da we will use triangle inea inequality 1203-10 12€. 12/24 equality. | Kean) telon)) - (Wit (₂) | = | an-dil + | b₁-d₂] 222 +2 Hence ... Kanton) | LE Convergent sequence is convergent. Sum of two
Transcribed Image Text:Explanation: counter example in Lan) = (-1) 9.) = 2-1, 41,-1,1,-1,-- y :nGN sequencence of real number which is not it is oscillatory. convergent it is oscillate finitely bu) Proof: - let two sequence g 30, I, and Is V given Exo Nyno be the points of Land and Lbn> respectively. I no G Z sif 1 our claim rs. we have Ny ho. Lan) - do I Le and 2 [an] tibn) cyf to dit da we will use triangle inea inequality 1203-10 12€. 12/24 equality. | Kean) telon)) - (Wit (₂) | = | an-dil + | b₁-d₂] 222 +2 Hence ... Kanton) | LE Convergent sequence is convergent. Sum of two
Notes- Definition:- gt sean, A sequence Lan) is convergent ift Such that for ESO for that. every 170-6124 Cauchy sequences - А seem (an) iff for such that, loe have Prooft- (C.): - let for some such that VE)0 nGN. have 1xm- and LE. there I there in NGN with n>N. Land is called Cauchy sequence there is NGA for every minGN. with min A, K₁, K₂ > max gronvergent to x, we 41,4230 , 3 n, 1₂ GA VnKishi and K₂) ₂ 1xK₁ -α1 24, and 19*2*1 242 80 both will hold for all is dєк such (n₁,1₂) E = max [E₁, (₂) then OK L ket). 10421-2-(2x₂-1) | LE 15x₁-xx²₂128 Proved Hence all convergent sequence is cauchy,
Transcribed Image Text:Notes- Definition:- gt sean, A sequence Lan) is convergent ift Such that for ESO for that. every 170-6124 Cauchy sequences - А seem (an) iff for such that, loe have Prooft- (C.): - let for some such that VE)0 nGN. have 1xm- and LE. there I there in NGN with n>N. Land is called Cauchy sequence there is NGA for every minGN. with min A, K₁, K₂ > max gronvergent to x, we 41,4230 , 3 n, 1₂ GA VnKishi and K₂) ₂ 1xK₁ -α1 24, and 19*2*1 242 80 both will hold for all is dєк such (n₁,1₂) E = max [E₁, (₂) then OK L ket). 10421-2-(2x₂-1) | LE 15x₁-xx²₂128 Proved Hence all convergent sequence is cauchy,
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