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Math and science::Analysis::Tao::06. Limits of sequences

The forward implication has been introduced already; it suffices to prove the reverse—that all Cauchy sequences are convergent.

The reverse is not a given: so far, for real sequences, we have delt with steadiness (Cauchy sequences) and closeness between a sequence and a real (convergence); however, we haven't introduced, until now, the idea that a steady sequence must converge to another real.

This theorem along with the closely reated idea of the existance of the supremum sets the reals apart from the rationals. In the language of metric spaces, the theorem asserts that the reals do not contain any 'holes'. The rationals in comparison, do contain holes—rationals can approach something that isn't a rational (and from both sides, hence the idea of a 'hole').

Note on equality

If the 'is' in the statment is considered 'is equal to' (which I think is an important distinction, as equality and entity can be different- two Cauchy sequences can be equal in that they represent the same real, yet they can contain different sequence values), then the Cauchy sequence here, I think, can be considered a Cauchy sequence of reals or rationals. If a sequence of reals converges to a real, then the real to which it converges can be expressed as a Cauchy sequence of rationals.

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Source

Tao, Analysis I
Chapter 6, p146