Density Of rationals

density-of-rationals

* Density of Real Numbers in R

  ∀a,b ∈ R, such that a < b, ∃t ∈ R such that a < t < b

** Proof
   ∀a, b, ∈ R, such that a < b

   let t = (a + b)/2 ∈ R since a, b ∈ R
   
   let δ = (b - a)/2 ∈ R
   since b > a, δ > 0

   t = (a + b) / 2 = a + (b - a) / 2 = a + δ
   
   => t = a + δ, where δ > 0
   therefore, t > a

   t = (b + a) / 2 = b - (b - a) / 2 = t - δ
   => t = b - δ, where δ > 0
   therefore, t < b

   Hence, a < t < b where t ∈ R