Complete Metric Spaces

Topology · Axiom Academy

Understanding Cauchy sequences and the fundamental property of completeness Before we can understand completeness, we need to introduce a special type of sequence that "wants" to converge. A sequence in a metric space is called a Cauchy sequence if for every , there exists an such that for all , In plain English: A sequence is Cauchy if its terms eventually get arbitrarily close to each other. As you go far enough out in the sequence, all subsequent terms cluster together within any desired distance. Step 2: Every Convergent Sequence is Cauchy Here's an important fact: if a sequence converges, it must be Cauchy. Let be a convergent sequence in a metric space . Then is a Cauchy sequence. Proof Sketch: Suppose converges to . Given , choose such that for all . Then for , we have: By the triangle inequality! So the sequence is Cauchy. Step 3: Complete Metric Spaces A metric space is "complete" if every Cauchy sequence actually converges to a point in the space. A metric space is called complete if every Cauchy sequence in converges to a limit in . In symbols: is complete if whenever is Cauchy, there exists such that . Step 4: The Real Numbers are Complete One of the most important examples of a complete metric space is with the standard metric . The real numbers with the usual metric form a complete metric space. What this means: Every Cauchy sequence of real numbers converges to a real number. There are no "missing limits" in ℝ.

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