Search results with tag "Complete metric"

Complete Metric Spaces

Complete Metric Spaces Definition 1. Let (X,d) be a metric space. A sequence (x n) in X is called a Cauchy sequence if for any ε > 0, there is an n ε ∈ N such that d(x m,x n) < ε for any m ≥ n ε, n ≥ n ε. Theorem 2. Any convergent sequence in a metric space is a Cauchy sequence. Proof. Assume that (x n) is a sequence which converges ...

  Complete, Metrics, Complete metric