Density of the Rationals and Irrationals in the Reals

Using the Well-Ordering Axiom and the Archimedean Property, we prove that both the rational and irrational numbers are dense in the real numbers.

0:00, Intro
0:05, Density
1:00, Well-Ordering Axiom
3:08, Lemma: Real numbers more than one apart have an integer in between
4:25, Proof of Lemma
9:02, Theorem: Q is dense in R
9:58, Proof of Q dense in R
12:55, Theorem: Set of irrationals is dense in R

[JoseLopez-opsq]

thank you, this helped clarify my confusion regarding how the inequality works out in the proof of rationals being dense in R. Now if I can only find a video that explains why the max{|nb|, |na|} is considered in alternate proof of density that would be great. When max {} or min{} are applied to proofs such as this it really confuses me. Anyway, thank you!