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Thm l1.2 Let (Csul be a (1) #(Sm.) sit Sue =s amolN ( Isu-si

Thm l1.2 Let (Csul be a (1) #(Sm.) sit Sue =s amolN ( Isu-si

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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Thm 1.2 Let Csul be a sequeuce. jif cut ouly it C tti (1) 3(Sm) sit lin Sie =s inelN ( Isu-sI <E} is arfinite Ve7o. k-00 Ar coly E70. (2) If Cu) s un baunded abase, then a subsnance (tu) st lu tu=. (3) If Csu) is unbaunded belaw , then a subsguance (tw) sit liu tu=- ∞. salzquarce In each case, the subsquance aun be taken to be mutonic. Pf. 1E We uce the cantradletan strategy. Assume thort there is a e7o st S= &uG Su-51<E} s fnre Chouse N= maxS. Then, n>N > So, 7(Sux) sit lim Suk = S. E Practce 1. Arsume u6INI ss Su< S+}is cntintte Vezo %3D = S. Shaw 3a decrousmey Subsequeme (Sua) Sit lim Sna = s.
Transcribed Image Text:Thm 1.2 Let Csul be a sequeuce. jif cut ouly it C tti (1) 3(Sm) sit lin Sie =s inelN ( Isu-sI <E} is arfinite Ve7o. k-00 Ar coly E70. (2) If Cu) s un baunded abase, then a subsnance (tu) st lu tu=. (3) If Csu) is unbaunded belaw , then a subsguance (tw) sit liu tu=- ∞. salzquarce In each case, the subsquance aun be taken to be mutonic. Pf. 1E We uce the cantradletan strategy. Assume thort there is a e7o st S= &uG Su-51<E} s fnre Chouse N= maxS. Then, n>N > So, 7(Sux) sit lim Suk = S. E Practce 1. Arsume u6INI ss Su< S+}is cntintte Vezo %3D = S. Shaw 3a decrousmey Subsequeme (Sua) Sit lim Sna = s.
Expert Solution
Step 1

Given Sn be a sequence.

(1).

To prove that ∃Snk such that limk→∞Snk=S⇔n∈ℕ| Sn-S<ε is infinite for all ε>0.

Forward Proof:

Suppose ∃Snk such that limk→∞Snk=S.

Assume by contradiction that there exists an ε>0 such that A=n∈ℕ| Sn-S<ε is a finite set.

Let N=max A, therefore for all n≥N, we have:

Sn-S≥ε.

Hence for all nk≥N, we have Snk-S≥ε, which is contradiction the fact that limk→∞Snk=S.

Hence our assumption must be wrong.

Hence the set A=n∈ℕ| Sn-S<ε is a finite set for all ε>0.

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