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In Strichartz's The Way of Analysis, Strichartz says that

"But the definition of Cauchy sequence does not involve the limit, as we wanted, so it is not immediately clear that every Cauchy sequence has a limit. Cauchy claimed to have proved this, but on a rigorous level his proof had to be bogus since he never defined 'number'"

Does this fact make the construction of real numbers using Cauchy sequences a little circular as we're essentially 'using the term in its own definition'. Also, what does this imply about the rigor of all of following lemmas that involve Cauchy sequences and the assumption that they have limits?

Dalop
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    If your metric space X is complete, then each Cauchy sequence has a limit. Otherwise, we don't know that a Cauchy sequence converges to a limit – Sean Roberson Dec 24 '19 at 23:40
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    There is absolutely no circularity in modern treatments of these topics. – Andrés E. Caicedo Dec 24 '19 at 23:49
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    For many centuries, the numbers were given by Nature, they didn't have to be defined. Did Cauchy think that it was necessary to define the real numbers? I haven't read his proof but he could have used the property that a non empty set of numbers bounded from above has a supremum for example. – Gribouillis Dec 25 '19 at 00:22
  • The construction of reals was not available to Cauchy and most certainly he couldn't have proved that every Cauchy sequence converges to a real number. Instead I believe Cauchy and his contemporaries assumed some principle of completeness and worked on the basis of it. Also there is no circularity in construction of reals because they are constructed as Cauchy sequences of rationals and not Cauchy sequences of reals. – Paramanand Singh Dec 25 '19 at 01:29
  • He didn't prove it. He only stated it and used it, and maybe, just maybe copied it from Bolzano. – MoonLightSyzygy Dec 25 '19 at 01:36

2 Answers2

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One is not using the term in its own definition, since one is defining real numbers as equivalence classes of Cauchy sequences, not of real numbers, but of rational numbers.

This may make it appear that rational numbers are not real numbers, and that is actually a limitation of the way of doing things that is called logical rigor.

Michael Hardy
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It seems we are considering "basic facts" about real numbers versus "basic assumptions" often called axioms. Before trying to prove cauchy sequence convergence we should identify what our basic assumptions are about $\Bbb R$. Analysis is based on the $\it{assumption}$, stated in one form or another, that $\Bbb R$ is the only complete ordered field.

Very often, depending on the text we are reading, we see basic facts presented as basic assumptions which then are used to prove old basic assumptions as if they were new basic facts. This "interchangeability" occurs because many of these so called facts or axioms are logically equivalent ; i.e. each one statement implies the other.

For example, the Monotone convergence "theorem" (MCT) is often presented as a basic fact in Analysis resting on the completeness axiom. However, we can simply assume MCT as an axiom and use it to prove the completeness "axiom" as a theorem.

Cauchy sequence convergence is a result of the completeness axiom but can also be assumed as an axiom and then used to prove the completeness "axiom" as a theorem. To prove basic assumptions or facts like Cauchy sequence convergence or the completeness axiom we must make assumptions beyond the scope of arithmetic. Even when we "construct" the reals using cauchy sequence equivalence classes we must make basic assumptions like the completeness axiom to prove cauchy sequence convergence.

Of course we cannot prove an assumption to be fact using only that very assumption again. That is, if cauchy sequence convergence is presented as a basic axiom of analysis then it shouldn't be provable and it is actually analysis itself that would result from that basic axiom.

Oliver Kayende
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