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Tag Archives: cauchy sequences
Topology: Complete Metric Spaces
[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchycontinuous maps. ] On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading
Posted in Notes
Tagged advanced, cauchy sequences, complete metric spaces, completion, metric spaces, topology, universal properties
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Topology: Cauchy Sequences and Uniform Continuity
[ Updated on 8 Mar 13 to include Cauchycontinuity and added answers to exercises. ] We wish to generalise the concept of Cauchy sequences to metric spaces. Recall that on an intuitive level, a Cauchy sequence is one where the … Continue reading
Posted in Notes
Tagged advanced, cauchy sequences, cauchycontinuity, metric spaces, product topology, topology, uniform continuity
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Basic Analysis: Sequence Convergence (2)
Monotone Convergence We start with a useful theorem. Monotone Convergence Theorem (MCT). A sequence is monotonically increasing (or just increasing) if for all n. Now the theorem says: an increasing sequence with an upper bound is convergent. Proof. Let L = sup{a1, a2, … }, … Continue reading