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Determine whether each of the following statements is true. If true, prove it. If false, give a counterexample. (a) Let an → o and b, → -0o. Then an – bn → 0. (b) Let a, +∞ and b, + oo. Then a,bn → 00. (c) Let an 0o and b, be unbounded above. Then an + ba + 00. (d) Let a, + 0 for all n, and a, → 0. Then 1/a, → 0.

Determine whether each of the following statements is true. If true, prove it. If false, give a counterexample. (a) Let an → o and b, → -0o. Then an – bn → 0. (b) Let a, +∞ and b, + oo. Then a,bn → 00. (c) Let an 0o and b, be unbounded above. Then an + ba + 00. (d) Let a, + 0 for all n, and a, → 0. Then 1/a, → 0.

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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Determine whether each of the following statements is true. If true, prove it. If false, give a counterexample. (a) Let an + 0o and b, + -00. Then an - bn → 00. (b) Let an + 0o and b, → 0o. Then a,b, + 0. (c) Let an + 0o and b, be unbounded above. Then an + bn → 00. (d) Let a, +0 for all n, and a, → 0. Then 1/an → 0. Remark: you must prove these results from first principles, that is, directly from the definitions. Do not try to deduce them from the Algebra of Limits!
Transcribed Image Text:Determine whether each of the following statements is true. If true, prove it. If false, give a counterexample. (a) Let an + 0o and b, + -00. Then an - bn → 00. (b) Let an + 0o and b, → 0o. Then a,b, + 0. (c) Let an + 0o and b, be unbounded above. Then an + bn → 00. (d) Let a, +0 for all n, and a, → 0. Then 1/an → 0. Remark: you must prove these results from first principles, that is, directly from the definitions. Do not try to deduce them from the Algebra of Limits!
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