Completeness of Metric Spaces

Key idea

Every Cauchy sequence converges.

In depth

A metric space is complete if every Cauchy sequence converges within it. R is complete, Q is not. Complete spaces are where fixed-point and existence theorems work.

Worked example

The sequence 3, 3.1, 3.14, ... is Cauchy in Q but converges to π, outside Q.

Real-world uses

• Ensuring that iterative computations on function spaces do not escape the space

Common mistake

Believing bounded closed subsets of any complete space are compact.

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.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• 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.