Density of Rationals in Reals

Real Analysis · Axiom Academy

LESSON Density of Rationals in Reals Between any two reals — no matter how close — there is always a fraction. We prove it, then watch it happen. A set is dense in the reals if it reaches into every gap: pick any two distinct reals and you can always find a member of the set strictly between them. The rationals pass this test — and the animation drops one fraction r into the gap between a and b , then keeps finding more. In words: for all reals there is a rational r with . The proof needs one tool: no positive real is "infinitely small." Take any target height x and any step size — stack enough steps and you will pass x . The animation stacks copies of until the tower clears the line at x . For every and every , some natural number n makes . 3. Proving the Density Theorem Given reals , we build a fraction between them. First make a grid of step finer than the gap b-a ; then the first grid point past a cannot overshoot b . The animation lays the grid down and walks to that point. By the Archimedean property pick with , so . Let m be the smallest integer with ; then . 4. The Fractions Never Run Out One fraction in the gap forces infinitely many: between a and sits another gap, so the theorem applies again, and again. The animation zooms into the sliver between the irrationals and — at every magnification, new fractions pop into view. apply density inside the new gap, forever and are not fractions — yet fractions crowd right up against them on both sides.

This is the written version of the interactive lesson above. See the full Real Analysis course.