**Density of Rational Numbers Theorem**
Given any two

real numbers α, β ∈

**R**, α<β, there is a

rational number r in

**Q** such that α<r<β.

**Proof:** Since α<β, β-α>0. 1>0 as well. We may use the

Archimedean property to conclude that there is some

integer m so 1 < m(β-α), or equivalently,

mα +1 < mβ.

Let n be the largest integer such that n ≤ mα. Adding 1 to both sides gives

n+1 ≤ mα +1 < mβ.

But since n is the largest integer less than or equal to mα, we know that mα < n+1 and therefore that

m α < n+1 < mβ or

α <(n+1)/m < β.

I like this

proof because it’s simplistic and low on

vocabulary; I suppose it’s more of a

hoi polloi-ish proof than the professors would prefer we use. The

Archimedean property is the most sophisticated tool you need to understand this, and there’s a good write-up on that. This proof is fantastic for someone being introduced to the study of

analysis or a non-major “stuck” taking a single semester of the stuff.

Taken from a homework assignment from a class titled "Fundamental Properties of Spaces and Functions: Part I" at the University of Iowa.