Completeness and Supremum

Key idea

Every nonempty bounded-above set has a least upper bound.

In depth

Every non-empty set bounded above has a least upper bound (supremum). This separates the reals from the rationals, which have gaps. The supremum need not belong to the set.

Worked example

sup{x : x² < 2} = √2, which does not exist in Q. sup(0, 1) = 1 but 1 is not in the set.

Formula

Real-world uses

• Explaining why a sequence of decimal approximations of √2 converges to a real number

Common mistake

Confusing the supremum with the maximum; a maximum must be in the set.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.
• Monotone Convergence Theorem – Bounded monotone sequences converge.