Completeness Consequences

Real Analysis · Axiom Academy

LESSON Completeness Consequences One axiom separates the reals from the rationals — and it is what guarantees that limits actually exist. Take . It is nonempty and bounded above — are all rational upper bounds. But each candidate is too big ( 1.5^2 = 2.25 ), so we shave it down. The bounds descend forever and never land on a smallest rational : the value they close in on is , which is not rational. A nonempty, bounded-above set of rationals …yet it has no least upper bound inside 2. Completeness Fills Every Cut The completeness axiom is the single property that distinguishes from . It says the descending upper bounds always have somewhere to land — so the same set S now has a supremum, and it is exactly . Every nonempty that is bounded above has a least upper bound . What makes the least upper bound is captured by two clauses — an upper bound, and one that nothing smaller can match: Every satisfies . Nothing in the set pokes above it. For every there is an with . Drop below M by any margin and the set still reaches into that gap. For viewed inside , we get . The supremum exists in even though it is not in S and not in — completeness only promises the bound exists, not that it is rational or attained. 3. The Payoff: Bounded + Increasing ⇒ Convergent

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