- #1

Jamin2112

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## 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?