(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm looking for a kick in the right direction of how to approach it from 1b onwards ('cos thankfully I can figure out 1a for myself). Please see the attached screenshot (I'm not good at making the formulas appear right).

2. Relevant equations

I'm really not sure. I've noticed looking back through the readings that the bit I've circled in b is one of the properties of the bisection process, e.g "the length of the interval is halved at each step". I'm wondering if this question is getting me to demonstrate that effect. Other thank that, this chapter covers intro to Bolzano-Weierstrass theorem, Cauchy sequences, and using these with the triangle inequality and a little bit of the telescoping property.

3. The attempt at a solution

Umm? I tried:

Let s_{k+2}- 1/2(s_{k+1}+ s_{k}) for all k[tex]\geq[/tex]1 and suppose |s_{k+2}- s_{k+1}|

then |s_{k+2}- s_{k+1}| = |1/2 (s_{k+1}- s_{k}) - 1/2 (s_{k}- s_{k-1})|

but given k[tex]\geq[/tex]1, isn't s_{k-1}invalid? If k=1 then s_{k-1}=s_{0}which isn't part of the sequence.

So I stopped.

then I thought how about then |s_{k+2}- s_{k+1}| = |1/2 (s_{k+1}- s_{k}) - s_{k+1}|

= |1/2(-s_{k+1}- s_{k})|

but then the s_{k+1}is negative, and I couldn't think of a way of making just that bit positive.

So I wondered if I was going about this completely the wrong way, and maybe I should be using the triangle inequality, except the answer isn't an inequality.

Hence I'm stumped as to how to start in on this thing, so if someone could give me a push, that would be really helpful.

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# Homework Help: Analysis - sequence proof

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