Cauchy sequences, induction, telescoping property

[Meggle]

Homework Statement

Scanned and attached

Homework Equations

I am guessing a combination of induction and the telescoping property.

The Attempt at a Solution

I'm studying this extramurally, and I've just hit a wall with this last chunk of the sequences section, so if someone can suggest a good textbook that would be really good too. My googlefu is not helping me this week.

Ok. So prove the first chunk of a). The only way I can think of to prove that chunk is by mathematical induction again, and I think from the fact that it doesn't explicitly say to use that that there ought to be another way. But I can't see it. So I think I've proved it using mathematical induction. Although that did involve saying |-$$\alpha$$(b-a)| = $$\alpha$$|(b-a)| which I'm a bit dubious about.

Second section of a) let n=1, then:
|s₁₊₁ - s₁| = |b - a| = $$\alpha$$ ⁰|b - a|
so the result holds for n=1.
Assume the result holds for some positive integer k, i.e. assume:
|s_k+1 - s_k| = $$\alpha$$^k-1|b - a|
so I think the first part shows the case where k=2 and $$\alpha$$^k-1=$$\alpha$$¹=$$\alpha$$ but I don't know how to make use of this.
??

So moving on to b) s_n+1 +$$\alpha$$s_n where n=1 becomes s₂ +$$\alpha$$s₁ = b + $$\alpha$$a ok for n=1. But I don't know how to prove the general, as the method my readings have relies on incorporating the definition of <s_n> into the equation to be proved, and I don't have a definition for s_k+1, or any examples in my readings of a proof involving s_k+2 instead of s_k+1.

Hence why I feel like I've just not got the jist of this section of the course material, but I've been back and forth through my readings and it's not becoming any clearer. Can anyone suggest the fundamental thinking I'm obviously not getting? Even some worked examples somewhere would be brilliant.

And I don't know why all my alphas are floating up so high, sorry.

 

[Raskolnikov]

You don't need to do induction on the first part of (a). It's just substitution of the formula they provide. If you get stuck at a spot, try going in the reverse direction and finding a spot where it meets the forward direction.

Although that did involve saying |-$$\alpha$$(b-a)| = $$\alpha$$|(b-a)| which I'm a bit dubious about.

That's perfectly fine here since they tell you $$\alpha$$ > 0.

For the 2nd part of (a), you were correct in assuming it works for k. Now you have to prove it works for k+1. To do this, you'll first have to substitute |s_(k+2) - s_(k+1)| with their respective values using the formula from the first part of (a). Next, after a bit of simplification and rearranging, you should be able to use your induction hypothesis to prove it works for k+1.