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4. Suppose that {s} is a bounded sequence and let s = sup{s:n € N}. Suppose also that Sn 0, we can always find some NEN such that -< & whenever n ≥ N and m≥ N). 72 m 6. Consider the sequence {x} given by x₂ = √n. (i) (ii) Show that for every & > 0, we can find some NEN such that Xn+1-Xn

4. Suppose that {s} is a bounded sequence and let s = sup{s:n € N}. Suppose also that Sn 0, we can always find some NEN such that -< & whenever n ≥ N and m≥ N). 72 m 6. Consider the sequence {x} given by x₂ = √n. (i) (ii) Show that for every & > 0, we can find some NEN such that Xn+1-Xn

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 82E
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4. Suppose that {s} is a bounded sequence and let s = sup{s:n € N}. Suppose also that Sn <s for every n E N (i.e the supremum is not a maximum). Show that {s} must have a strictly increasing subsequence that converges to s. (Hint: For any particular k € N, s − is NOT an upper bound for {n}). k 5. Show directly from the definition that the sequence satisfies the Cauchy criterion. (i.e You may not use the fact that converges. You need to show how given any e > 0, we can always find some N E N such that |-< & whenever In n ≥ N and m≥ N). 6. Consider the sequence {x} given by x₂ = √n. (i) (ii) Show that for every & > 0, we can find some NEN such that Xn+1-Xn < & whenever n > N. Show that {x} is NOT a Cauchy sequence. (This example shows that the condition in (i) is not as strong as the Cauchy criterion).
Transcribed Image Text:4. Suppose that {s} is a bounded sequence and let s = sup{s:n € N}. Suppose also that Sn <s for every n E N (i.e the supremum is not a maximum). Show that {s} must have a strictly increasing subsequence that converges to s. (Hint: For any particular k € N, s − is NOT an upper bound for {n}). k 5. Show directly from the definition that the sequence satisfies the Cauchy criterion. (i.e You may not use the fact that converges. You need to show how given any e > 0, we can always find some N E N such that |-< & whenever In n ≥ N and m≥ N). 6. Consider the sequence {x} given by x₂ = √n. (i) (ii) Show that for every & > 0, we can find some NEN such that Xn+1-Xn < & whenever n > N. Show that {x} is NOT a Cauchy sequence. (This example shows that the condition in (i) is not as strong as the Cauchy criterion).
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