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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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