Dense Subsets

Real Analysis · Axiom Academy

Prove that the rationals are dense in the reals — every real can be approximated arbitrarily closely by a rational. A subset D of a metric space X is dense if its closure is all of X : . Equivalently, every nonempty open ball contains a point of D . Show that is dense in — that is, for every real x and every there is a rational q with , so . No matter how tiny you make the window around a real x , a rational q is caught inside it. That is exactly what "dense" means — and why . Nicely done — you built a rational inside any window around a real, which is exactly density. Dense means approximable: for any real x and any precision , some rational sits within of x , so . The Archimedean property is the engine: it lets us pick n with , making the grid as fine as needed. Construction, not magic: the least integer gives q = m/n with , forcing . Between any two reals lies a rational (and an irrational) — the same idea shows the irrationals and the dyadic rationals are dense in too. Why it matters: a continuous function is determined by its values on a dense set, and (Weierstrass) the polynomials are dense in C[a,b] . Even though is countable and is not, the rationals are "everywhere" in the reals.

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