Completeness Theorem of the Real Numbers

analysis

Definition

Completeness of the Real Numbers

Every non-empty subset of the real numbers $\mathbb{R}$ that is bounded above has a supremum in $\mathbb{R}$.

Similarly, every non-empty subset of $\mathbb{R}$ that is bounded below has an infimum in $\mathbb{R}$.

This property distinguish the real numbers from the rational numbers $\mathbb{Q}$, which are “incomplete” (e.g., the set $\{x\in \mathbb{Q}\mid x^2<2\}$ has no supremum in $\mathbb{Q}$).

Equivalent Formulations

The completeness of $\mathbb{R}$ is equivalent to several other fundamental principles in analysis:

• Cauchy Completeness: Every Cauchy sequence of real numbers is a convergent sequence.
• Nested Interval Property: For every sequence of nested closed intervals $[a_n,b_n]$ whose lengths approach zero, there is exactly one point contained in all intervals.
• Bolzano–Weierstrass Theorem: Every bounded sequence has a convergent subsequence.
• Monotone Convergence Theorem: Every bounded monotone sequence converges.