The Completeness Axiom: Suprema and Infima
The rationals have gaps and the reals do not. Completeness is stated as every bounded set having a least upper bound, and it is the axiom every later theorem in analysis is traced back to.
What a learner can do afterwards
- Find the supremum of a set that has no maximum
- Show that the rationals fail the least-upper-bound property
- Use the supremum in an argument rather than the maximum
1 · Read
An upper bound sits at or above every member of a set, and the supremum is the least such bound. The set {1 minus 1 over n} climbs forever with no largest term, yet its supremum is 1. The open interval (0, 1) likewise has supremum 1 but no maximum. Mirroring downward, the greatest lower bound is the infimum, so inf {1 over n} is 0.
The rationals have gaps the reals do not. The set of rationals q with q squared below 2 is bounded above, but it has no rational least upper bound, because root 2 is missing from the rationals. That failure is exactly what completeness fixes.
The completeness axiom says every nonempty bounded above set of reals has a real supremum. Use the supremum in an argument whenever the maximum may not exist. Bounds come first and N second: to get within 0.001 of the supremum 1, the gap 1 over n below 0.001 needs n 1001 or more, so the smallest whole n is 1001.
The supremum is the least ceiling even when no top step exists, and completeness guarantees the reals always supply one.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.