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.

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