Let {an}=1 be a sequence of real numbers that converges to A. Using only the definition of convergence and Theorems from Section 1.1 or before, prove that {a} converges to A², by following the series of steps outlined below. Your job is to complete the scratchwork as scaffolded and then write a proof of the theorem. n=1 (a) Scratchwork step 1: Explain why there exists M > 0 such that |an| < M for all n E J. (b) Step 2: Show the algebra required to justify why |a – A²| < ]an – A|(M +|A|) (c) Step 3: Explain why there exists N E J such that |an – A| < for all n > N. M +|A| (d) Proof: Let e > 0 be given. Complete the rest of the proof explicitly referring to the scratchwork above.

Let {an}=1 be a sequence of real numbers that converges to A. Using only the definition of convergence and Theorems from Section 1.1 or before, prove that {a} converges to A², by following the series of steps outlined below. Your job is to complete the scratchwork as scaffolded and then write a proof of the theorem. n=1 (a) Scratchwork step 1: Explain why there exists M > 0 such that |an| < M for all n E J. (b) Step 2: Show the algebra required to justify why |a – A²| < ]an – A|(M +|A|) (c) Step 3: Explain why there exists N E J such that |an – A| < for all n > N. M +|A| (d) Proof: Let e > 0 be given. Complete the rest of the proof explicitly referring to the scratchwork above.

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Description: Let {an}=1 be a sequence of real numbers that converges to A. Using only the definition of convergence andTheorems from Section 1.1 or before, prove that {a} converges to A², by following the series of stepsoutlined below. Your job is to complete th

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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Let {an}=1 be a sequence of real numbers that converges to A. Using only the definition of convergence and
Theorems from Section 1.1 or before, prove that {a} converges to A², by following the series of steps
outlined below. Your job is to complete the scratchwork as scaffolded and then write a proof of the theorem.
n=1
(a) Scratchwork step 1: Explain why there exists M > 0 such that |an| < M for all n E J.
(b) Step 2: Show the algebra required to justify why |a – A²| < ]an – A|(M +|A|)
(c) Step 3: Explain why there exists N E J such that
|an – A| <
for all n > N.
M +|A|
(d) Proof: Let e > 0 be given. Complete the rest of the proof explicitly referring to the scratchwork above.

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