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Cauchy and sequences

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Respected Sir / Madam,

I need following question to be answered.

Please mention each and every step. Please explain every thing.
Text Book:- Taylor & Menon.
Page 523 Exercise Problem.

Page 526 Exercise Problem

Thanking you

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Solution Summary

This shows how to prove that Cauchy's condition is fulfilled and complete a proof regarding a set and intervals.

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Please see the attachment.

Page 523
Proof:
Since , and for , then we have

Then we get the absolute value on both sides and get

In general, we have

Now we consider and without generality, we assume , where is a nonnegative integer. Then we have

...

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