2. Intervals in R

2.1. Density of Q in R

Statement:

Between any two real numbers

a<ba < b

, there exists a rational number

qq

q such that

a<q<ba < q < b

Proof Sketch:

Let

a<ba < b

. Since

b−a>0b - a > 0

, choose

n∈Nn \in ℕ

such that

1n<b−a\frac{1}{n} < b - a

. Then choose

m∈Zm \in ℤ

such that

mn>a\frac{m}{n} > a

. Adjust

mm

m to find a rational number

q=mn∈(a,b)q = \frac{m}{n} \in (a, b)

Implication:

  • The rationals are “everywhere” in the reals

  • Rational approximations are always possible

  • Fundamental in defining limits and continuity