Tag Archives: cauchy sequences

Topology: Complete Metric Spaces

Posted on February 21, 2013 by limsup

[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchy-continuous maps. ]  On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading →

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Topology: Cauchy Sequences and Uniform Continuity

Posted on February 7, 2013 by limsup

 [ Updated on 8 Mar 13 to include Cauchy-continuity 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 →

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Basic Analysis: Sequence Convergence (2)

Posted on November 12, 2012 by limsup

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 →

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