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(ii) If (sn) is an unbounded decreasing sequence, then lim så -∞

(ii) If (sn) is an unbounded decreasing sequence, then lim så -∞

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove ii. the unbounded decreasing sequences,

10.4 Theorem. (i) If (sn) is an unbounded increasing sequence, then lim sn +∞. (ii) If (sn) is an unbounded decreasing sequence, then lim sn ·∞. Proof (i) Let (sn) be an unbounded increasing sequence. Let M > 0. Since the set {sn : n € N} is unbounded and it is bounded below by $₁, it must be unbounded above. Hence for some N in N we have sã > M. Clearly n > N implies sn ≥ sn > M, so lim Sn = +∞o. sn (ii) The proof is similar and is left to Exercise 10.5.
Transcribed Image Text:10.4 Theorem. (i) If (sn) is an unbounded increasing sequence, then lim sn +∞. (ii) If (sn) is an unbounded decreasing sequence, then lim sn ·∞. Proof (i) Let (sn) be an unbounded increasing sequence. Let M > 0. Since the set {sn : n € N} is unbounded and it is bounded below by $₁, it must be unbounded above. Hence for some N in N we have sã > M. Clearly n > N implies sn ≥ sn > M, so lim Sn = +∞o. sn (ii) The proof is similar and is left to Exercise 10.5.
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