Question: Let {an} be a sequence and L is any real number. Prove that {an} =1 converges to L if and only if every monotone subsequence of {an} converges to L . Tip: prove the contrapositive by using a) in image, and the Bolzano-Weierstrass Theorem. (a) Let {an} be any sequence and L any real number. Prove that if {a,} doesnot converge to L, then there exists an e> 0 and a subsequence {a} suchthat an - L > e for all k.Tip: Start by writing down what it means that {am} does not converge to L,i.e. write down the formal negation of the statement {a,} converges to L. Usethis formal negation and induction on k to construct an increasing sequence ofnatural numbers {ng} such that |an - L| > e for all k.3D1

Name: Question: Let {an} be a sequence and L is any real number. Prove that
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Description: Question: Let {an} be a sequence and L is any real number. Prove that {an} =1 converges to L if and only if every monotone subsequence of {an} converges to L . Tip: prove the contrapositive by using a) in image, and the Bolzano-Weierstrass Theorem

Consider a sequence, an and L be any real number. To prove that, an converges to L if and only if…

Answered: Question: Let {an} be a sequence and L…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...

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Question: Let {an} be a sequence and L is any real number. Prove that {an} =1 converges to L if and only if every monotone subsequence of {an} converges to L .

Tip: prove the contrapositive by using a) in image, and the Bolzano-Weierstrass Theorem.

(a) Let {an} be any sequence and L any real number. Prove that if {a,} does
not converge to L, then there exists an e> 0 and a subsequence {a} such
that an - L > e for all k.
Tip: Start by writing down what it means that {am} does not converge to L,
i.e. write down the formal negation of the statement {a,} converges to L. Use
this formal negation and induction on k to construct an increasing sequence of
natural numbers {ng} such that |an - L| > e for all k.
3D1

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