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Cauchy Sequences
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Cauchy sequences are an important definition in calculus and real analysis.
In this video we begin by defining what a cauchy sequence is.
Following this we prove that all convergent sequences are cauchy.
Then we look at how in the real line all Cauchy sequences are convergent.
We see how this follows from the Bolzano-Weierstrass theorem.
Following this we look at an example of a cauchy sequence in the rationals which does not converge to a limit in the rationals. We do this to demonstrate how it is not true that all cauchy sequences converge in the rationals.
Finally we finish by discussing the equivalence of the LUB axiom and cauchy completeness to characterise the completeness axiom of the real line.
In this video we begin by defining what a cauchy sequence is.
Following this we prove that all convergent sequences are cauchy.
Then we look at how in the real line all Cauchy sequences are convergent.
We see how this follows from the Bolzano-Weierstrass theorem.
Following this we look at an example of a cauchy sequence in the rationals which does not converge to a limit in the rationals. We do this to demonstrate how it is not true that all cauchy sequences converge in the rationals.
Finally we finish by discussing the equivalence of the LUB axiom and cauchy completeness to characterise the completeness axiom of the real line.
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