# Introduction to Real Analysis (4th Edition) View more editions 93% (60 ratings) for Chapter 4.3Solutions for Chapter 4.3

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Chapter: Problem:
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Chapter: Problem:
• Step 1 of 2

Let

Let

Let be a cluster point of

Now we will prove that the following statements are equivalent:

(i)

(ii) For every sequence in such that

We have

:

For any

There exists such that for all with, then

Since then we can

Choose such that for any we have

From the definition of the right hand limit of f at c we have,

.

Therefore we have

Thus the sequenceconverges to L.

• Step 2 of 2

:

If is not true.

Since is a cluster point of there exist

For any there exists such that

, and

Then we find a sequence such that

From if exists then the sequencenot converges to L.

This is a contradiction to the statement

And the assertion is not true.

So (ii) is not true.

Then we can conclude that if (ii) is true, thenis true.

Hence the two statements are equivalent.

Corresponding Textbook

Introduction to Real Analysis | 4th Edition
9781118135853ISBN-13: 1118135857ISBN:
This is an alternate ISBN. View the primary ISBN for: Introduction to Real Analysis 4th Edition Textbook Solutions
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