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The Real Numbers Have No Holes

Limits live or die on what numbers exist to land on. The rationals have invisible gaps — a sequence can crowd toward a point that isn't there. The fix is one decisive axiom, [[completeness-axiom|completeness]], which fills every hole. We prove a gap exists, then see how the [[least-upper-bound-property|least upper bound property]] seals it.

A hole you can prove is there

The rational numbers feel complete — between any two of them lies another, forever; this density makes them look gapless. But looks deceive again. Consider the set of positive rationals whose square is less than 2. It is bounded above (by 2, say), so it ought to have a least upper bound — a smallest ceiling. Among the rationals, that ceiling would have to be a number whose square is exactly 2. And no such rational exists.

Claim:  there is NO rational number x with x^2 = 2.

Proof by contradiction.
  Suppose x = p/q is rational, in LOWEST terms (p, q share no factor),
  and x^2 = 2.   Then  p^2 / q^2 = 2,  so  p^2 = 2 q^2.

  So p^2 is even.  A square is even only if its root is even,
  so p is even:  write  p = 2k.

  Then  (2k)^2 = 2 q^2  ->  4 k^2 = 2 q^2  ->  q^2 = 2 k^2.
  So q^2 is even, hence q is even too.

  But now p and q are BOTH even -- they share the factor 2.
  That contradicts "lowest terms."

  The assumption must be false.  No rational squares to 2.    QED.

Consequence: the set  A = { x in Q : x > 0 and x^2 < 2 }  is bounded
above but has NO least upper bound INSIDE the rationals. There is a
HOLE in Q exactly where sqrt(2) should be.
The classic proof that √2 is irrational, repurposed: it exhibits a bounded set of rationals with no rational least upper bound — a genuine hole in the rational line.

The axiom that fills every hole

The real numbers are built precisely so this never happens. The defining law is the completeness axiom, usually stated as the least upper bound property: every nonempty set of reals that has an upper bound has a *least* one, called its supremum. There is no “smallest ceiling that is missing.” Every set that should have a top has its top, sitting right there in the reals.

Apply it to our set A = { x > 0 : x² < 2 }. Over the reals, A has an upper bound, so completeness *hands us* a real number s = sup A. One then proves s² is neither less than 2 nor greater than 2 (each would let you nudge s and contradict its being the least upper bound), so s² = 2. The completeness axiom literally manufactures √2 as the supremum of a set that pointed at an empty spot. This is how the continuum is made hole-free.

Why limits need completeness

Completeness is not abstract bookkeeping; it is what makes limits *land*. Take the rational decimal truncations of √2: 1, 1.4, 1.41, 1.414, ... This sequence is increasing and bounded above by 1.5. Over the rationals it converges to nothing — its target is a hole. Over the reals, completeness guarantees the target exists: every bounded increasing sequence converges, and it converges to the supremum of its values.

This is the quiet engine under nearly every existence theorem you will meet. A root exists *because* the reals are complete; a sequence that bunches up — a Cauchy sequence — has a limit *because* the reals are complete; the Bolzano–Weierstrass and nested interval theorems are all completeness wearing different clothes. Take completeness away and analysis collapses; put it in, and limits finally have somewhere to land.