- #1

cragar

- 2,552

- 3

## Homework Statement

Prove that the dyadic rationals are dense in Q.

That is the rationals of the form [itex] \frac{m}{2^n} [/itex]

m is an integer and n is a natural

## The Attempt at a Solution

Let's say we have two arbitrary rationals x and y. where x<y

Now I will pick a rational smaller than x such that it is of the form

[itex] \frac{s}{2^k} [/itex] and i will call this P ,

now I will pick a rational larger than y that is of the same form

and i will call it O .

Now I will add P and O together and then divide by 2, find the midpoint

Now this new rational has a denominator that is a power of 2 because

everything we did had a denominator of 2. Now I will keep doing this,

I will keep finding mid points between these sets of rationals

that I created and I might have to pick the left or right one and then

keep finding the midpoints. Eventually i will get in between x and y.

I realize this is informal but Is my general idea in the right direction.