3: Vector Spaces and Metric Spaces
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- 3.13: Cauchy Sequences. Completeness
- A convergent sequence is characterized by the fact that its terms xₘ become (and stay) arbitrarily close to its limit, as m→+∞. Due to this, however, they also get close to each other; in fact, ρ(xₘ,xₙ) can be made arbitrarily small for sufficiently large m and n. It is natural to ask whether the latter property, in turn, implies the existence of a limit. This problem was first studied by Augustin-Louis Cauchy (1789−1857). Thus we shall call sequences Cauchy sequences.