Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many s belong to [a, b], then lim s belongs to [a, b].
Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many s belong to [a, b], then lim s belongs to [a, b].
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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![8.9
² Let (sn) be a sequence that converges.
(a) Show that if så ≥ a for all but finitely many n, then lim sɲ ≥ a.
'n
(b) Show that if så ≤ b for all but finitely many n, then lim så ≤ b.
(c) Conclude that if all but finitely many s belong to [a, b], then
lim sn belongs to [a, b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833bf7b1-3e6b-4749-8e88-54090320a3f5%2F4d483534-a080-4423-9cd3-d5fa36c1496f%2Faekj2yg_processed.png&w=3840&q=75)
Transcribed Image Text:8.9
² Let (sn) be a sequence that converges.
(a) Show that if så ≥ a for all but finitely many n, then lim sɲ ≥ a.
'n
(b) Show that if så ≤ b for all but finitely many n, then lim så ≤ b.
(c) Conclude that if all but finitely many s belong to [a, b], then
lim sn belongs to [a, b].
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