If a and b are rational numbers satisfying

[Jamin2112]

Homework Statement

Prove Proposition 1.15.

Proposition 1.15. If a and b are real numbers satisfying a<b, then there are rational numbers and irrational numbers between a and b.

Homework Equations

Professor said to use the Archimedean property

The Attempt at a Solution

a < b ---> (a+a)/2 < (a+b)/2 ---> a < (a+b)/2

Similarly,

a < b ---> (a+b)/2 < (b+b)/2 ---> (a+b)/2 < b

Hence a < (a+b)/2 < b

Now I need some way to show that (a+b)/2 can equal rational and irrational numbers. Any suggestions?

 

[tiny-tim]

Hi Jamin2112!

But you haven't used the Archimedean property.

Hint: try it with a = 0. ​