Courant's Fundamentals, help with proofs

[DrWillVKN]

Homework Statement

a) For any fixed integer q > 1, prove that the set of points x = p/q^8, p, s ranging over all positive integers, is dense on the number line

b) Show that if p is required to range only over a finite interval, p<= M for some fixed M, the set of all x is not dense on any interval

Homework Equations

n/a

The Attempt at a Solution

?
I have trouble understanding the problems presented in this book. What are x, p and s? What do you even do with them? Rational points are dense if they are arbitrarily close, which can be done by making q a large positive integer. So those set of points would have to be arbitrarily close to a point P on the number line.

A finite interval would not be dense. But then what?

I took a proof based class, but the problems covered used material taught in the corresponding chapter, making it easy to pick the techniques and theorems to proof the problems. Here, I have a horrible time understanding the problems and what to use to solve them.

I have looked at spivak's book and the first few chapters were fine, but then the problems started getting as confusing as the problems here. I am afraid of using an 'easier' book because 1) I don't think there is an easier book than these (spivak, courant) and 2) A book easier than this would not be as rigorous, and I'm not sure if it'd help me understand these problems, which I will have to deal with anyways when I take upper level math courses.

 

[Nino MMP]

I was thinking of taking a course at my school, but the courses here also use the same books as the ones I'm reading now. Does anyone have any idea what I should do?