Number systems: construction of rational numbers

[superfloyd]

Homework Statement

let a and b be rational numbers
if a>b>0
show that there exists a natural number n, which can be considered rational as well, st
nb>a

Homework Equations

The Attempt at a Solution

I was trying using the peano theorem of natural numbers

define set of natural numbers st n E S.
Assume n =1
(1)b = b
we know that b <a. hence 1 E N.

Assume n E n. then n' E n
n'b >a
(n+1)b
bn +b
b(1) + b
2b

I am kinda confused need some help in this thanks very much

 

[mighty2000]

Thank you for your post. It seems like you are on the right track with using the Peano theorem to solve this problem. However, I believe there are some steps missing in your attempt.

Firstly, you have defined a set of natural numbers but it is not clear how this set is related to the problem at hand. Instead, you could try to define a natural number n in terms of a and b, such that nb>a. One way to do this is to consider the fraction a/b. Since a and b are both rational numbers, we know that a/b is also a rational number. Now, we can use the fact that a>b to show that a/b is greater than 1. This means that we can find a natural number n such that n>a/b. Then, we can multiply both sides by b to get nb>a.

I hope this helps. Let me know if you need further clarification. Good luck with your solution!