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Problem 7.4.1. Prove that if limn0 Sn = s then lim,,→Sn| = |s|. %3D Prove that the converse is true when s = 0, but it is not necessarily true otherwise. Problem 7.4.2. (a) Let (sn) and (tn) be sequences with 8n < tn, Vn. Suppose lim,, 8n lim,00 tn = t. = s and Prove s < t. Hint. Assume for contradiction, that s >t and use the definition of * to produce an convergence with e = n with sn > tn. (b) Prove that if a sequence converges, then its limit is unique. That is, prove that = t, then if limn→0 Sn = s and limn→∞ Sn

Problem 7.4.1. Prove that if limn0 Sn = s then lim,,→Sn| = |s|. %3D Prove that the converse is true when s = 0, but it is not necessarily true otherwise. Problem 7.4.2. (a) Let (sn) and (tn) be sequences with 8n < tn, Vn. Suppose lim,, 8n lim,00 tn = t. = s and Prove s < t. Hint. Assume for contradiction, that s >t and use the definition of * to produce an convergence with e = n with sn > tn. (b) Prove that if a sequence converges, then its limit is unique. That is, prove that = t, then if limn→0 Sn = s and limn→∞ Sn

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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Real math analysis. Please Help solve this Problem.

Divergence.

10:59 O 1.4 AUUTIONa lTTODTEHIS Problem 7.4.1. Prove that if lim,00 Sn = s then limn-→ 8n = |s|. Prove that the converse is true when s = 0, but it is not necessarily true otherwise. Problem 7.4.2. (a) Let (sn) and (tn) be sequences with Sn < tn, Vn. Suppose limn0 8n = s and lim, +0 tn = t. Prove s < t. Hint. Assume for contradiction, that s >t and use the definition of * to produce an convergence with ɛ = n with sn > tn. (b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn = s and limn→∞ Sn t, then s = t. Problem 7.4.3. Prove that if the sequence (sn) is bounded then lim, 00 () = 0. n→∞∞ n II II
Transcribed Image Text:10:59 O 1.4 AUUTIONa lTTODTEHIS Problem 7.4.1. Prove that if lim,00 Sn = s then limn-→ 8n = |s|. Prove that the converse is true when s = 0, but it is not necessarily true otherwise. Problem 7.4.2. (a) Let (sn) and (tn) be sequences with Sn < tn, Vn. Suppose limn0 8n = s and lim, +0 tn = t. Prove s < t. Hint. Assume for contradiction, that s >t and use the definition of * to produce an convergence with ɛ = n with sn > tn. (b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn = s and limn→∞ Sn t, then s = t. Problem 7.4.3. Prove that if the sequence (sn) is bounded then lim, 00 () = 0. n→∞∞ n II II
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