Thm 1.2 Let Csul be a sequeuce. jif cut ouly it C tti(1) 3(Sm) sit lin Sie =s inelN ( Isu-sI 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.

Given Sn be a sequence. (1). To prove that ∃Snk such that limk→∞Snk=S⇔n∈ℕ| Sn-S<ε is infinite for…

Answered: Thm l1.2 Let (Csul be a (1) #(Sm.) sit…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.6: Quadratic Functions

Problem 38E

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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.

Expert Solution

Step 1

Given $(S_{n})$ be a sequence.

(1).

To prove that $\exists S_{n_{k}}$ such that $\lim_{k \to \infty} S_{n_{k}} = S \Leftrightarrow \{n \in ℕ | |S_{n} - S| < ε\}$ is infinite for all $ε > 0$ .

Forward Proof:

Suppose $\exists S_{n_{k}}$ such that $\lim_{k \to \infty} S_{n_{k}} = S$ .

Assume by contradiction that there exists an $ε > 0$ such that $A = \{n \in ℕ | |S_{n} - S| < ε\}$ is a finite set.

Let $N = max (A)$ , therefore for all $n \geq N$ , we have:

$|S_{n} - S| \geq ε$ .

Hence for all $n_{k} \geq N$ , we have $|S_{n_{k}} - S| \geq ε$ , which is contradiction the fact that $\lim_{k \to \infty} S_{n_{k}} = S$ .

Hence our assumption must be wrong.

Hence the set $A = \{n \in ℕ | |S_{n} - S| < ε\}$ is a finite set for all $ε > 0$ .

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