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(b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn = s and limn→∞ $n = t, then s = t.

(b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn = s and limn→∞ $n = t, then s = t.

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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Can I please get help solving the second part.

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
8:11 よ 回 bartleby.com/questions-and SEARCH &, ASK Vx MATH SOLVER CHAT Problem 7.4. : Let lim Sn = real numbon. natural rurnber Let E>o be any there exist a Then such that Vnとと We know Sn - > lim n→の we will show conver se is true when s=g Let lim Snl = s = 0 such that Then for any E>o, MEIN Vnz m Step 3 > | Sn| < € vnzm [sir Since > lim Sn=0 n→の is true when is not necessarily true. Example:- i.converse othercwise Let us consider | sn/ = 1G)") = -iM=(4)^= 1 > lim Sn|= 1 Then n-→の But San= EDan = (-)"=0)"= / II
Transcribed Image Text:8:11 よ 回 bartleby.com/questions-and SEARCH &, ASK Vx MATH SOLVER CHAT Problem 7.4. : Let lim Sn = real numbon. natural rurnber Let E>o be any there exist a Then such that Vnとと We know Sn - > lim n→の we will show conver se is true when s=g Let lim Snl = s = 0 such that Then for any E>o, MEIN Vnz m Step 3 > | Sn| < € vnzm [sir Since > lim Sn=0 n→の is true when is not necessarily true. Example:- i.converse othercwise Let us consider | sn/ = 1G)") = -iM=(4)^= 1 > lim Sn|= 1 Then n-→の But San= EDan = (-)"=0)"= / II
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 sn such that limn→∞ sn=s and limn→∞ sn=t, for all natural

numbers n∈ℕ.

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