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Problem 7.4.2. (a) Let (8n) and (t,) be sequences with Sn < tn, Vn. Suppose lim,n0 8n = s and lim,00 tn = t. S Prove s < t. Hint. Assume for contradiction, that S s >t and use the definition of s-t * to produce an convergence with e = n with sn > tn. 2

Problem 7.4.2. (a) Let (8n) and (t,) be sequences with Sn < tn, Vn. Suppose lim,n0 8n = s and lim,00 tn = t. S Prove s < t. Hint. Assume for contradiction, that S s >t and use the definition of s-t * to produce an convergence with e = n with sn > tn. 2

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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I need help with 7.4.2 part (a) please

10:33 dD bartleby.com/questions-and VIIZ II Jtep 5 S tvIIZITI [SIT Smce M SN-U I when is not necessarily true. Example:- i.converse othe us consider | sn/= 1G)") = -iM=(4)^=1> lim Sn|= 1 Then San= EDan = (-)"=0)"= / | Expert Solution Step 1 We have to give the proof of the second part i.e., part (b) of problem 7.4.2. We have to show that if a sequence is convergent then its limit is unique. Let us consider the sequence (s,) such that lim s, = sand lim S, = t, for all natural n-00 n-00 numbers n E N. Step 2 We have lim Sn = s and lim s, = t. This implies that the sequence (s,) tends to different limits s n-00 and t. We can write lim S, - lim s, = s - t. Now, lim S. lim S=s -t n-00 lim (S, - s,)=s - t lim (0)=s -t n- 00 0=s - t (: limit of a constant sequence containing 0 is 0) s=t Therefore, the limit of a convergent sequence is unique. Got Rate this solution 7 Students who've seen this question also like: Advanced Engineering Mathematics Systems Of Odes. Phase Plane. Qualitative Methods. 1RQ Stuck on vour II
Transcribed Image Text:10:33 dD bartleby.com/questions-and VIIZ II Jtep 5 S tvIIZITI [SIT Smce M SN-U I when is not necessarily true. Example:- i.converse othe us consider | sn/= 1G)") = -iM=(4)^=1> lim Sn|= 1 Then San= EDan = (-)"=0)"= / | Expert Solution Step 1 We have to give the proof of the second part i.e., part (b) of problem 7.4.2. We have to show that if a sequence is convergent then its limit is unique. Let us consider the sequence (s,) such that lim s, = sand lim S, = t, for all natural n-00 n-00 numbers n E N. Step 2 We have lim Sn = s and lim s, = t. This implies that the sequence (s,) tends to different limits s n-00 and t. We can write lim S, - lim s, = s - t. Now, lim S. lim S=s -t n-00 lim (S, - s,)=s - t lim (0)=s -t n- 00 0=s - t (: limit of a constant sequence containing 0 is 0) s=t Therefore, the limit of a convergent sequence is unique. Got Rate this solution 7 Students who've seen this question also like: Advanced Engineering Mathematics Systems Of Odes. Phase Plane. Qualitative Methods. 1RQ Stuck on vour II
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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