Density of Rational Numbers Theorem
Given any two real number
s α, β ∈ R
, α<β, there is a rational number
r in Q
such that α<r<β.
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.