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I was reading Sadri Hassani in which he mentioned that the rational numbers by themselves are an incomplete metric. He showed this by constructing a Cauchy sequence whose limit point was $\log2$. He then said that to complete the metric space, you would need the whole of real numbers. My question is if can you prove that there exists a Cauchy sequence whose limit point is an arbitrary real number? That would justify the need of the whole real line.

Andrés E. Caicedo
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Rishabh Jain
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    One construction of the real numbers is as all equivalence classes of Cauchy sequences of rational numbers. In that construction of the real numbers you can just take a representative of each real number. In other words, the real line is defined as all numbers that are needed. – conditionalMethod Oct 21 '19 at 20:15
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    How about, say, $3, 3.1, 3.14, 3.141, 3.1415, 3.14159, \dots$ for $\pi$? – 79037662 Oct 21 '19 at 20:15
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    Write a given real number as an (infinite) decimal, then approximate it by cutting off the decimal representation at increasingly longer points. – rogerl Oct 21 '19 at 20:15
  • @79037662 Got it , thanks – Rishabh Jain Oct 21 '19 at 20:22
  • As conditionalMethod says, the usual definition of the real numbers is basically "what you need to complete $\mathbb{Q}$." So to make this nontrivial we need to rephrase it somehow. One approach would be to focus on a different definition of the reals (say, Dedekind cuts); another is to ask a more "localized" question, like how we know that there is no countable completion of $\mathbb{Q}$ (basically, why $\mathbb{R}$ is uncountable). Certainly the latter question has been asked here several times; I believe the former has been as well. But at present your question is "potentially trivial." – Noah Schweber Oct 21 '19 at 21:07
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    @rogerl that begs the question: why do real numbers have such a decimal expansion ? It must follow from the definition of the real numbers. – Henno Brandsma Oct 21 '19 at 21:18
  • The decimal expansion is just a convergent series. – nguyen quang do Oct 23 '19 at 09:28
  • @HennoBrandsma See this question https://math.stackexchange.com/questions/3385949/going-from-completion-to-explicit-description-of-the-real-numbers#comment6966314_3385949 – 79037662 Oct 23 '19 at 14:06

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