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The real numbers can be constructed as the set of Cauchy sequences (e.g. 3, 3.1, 3.14, 3.141, 3.1415,...) under the equivalence relation that their difference tends to 0.

  • Real numbers are equivalence classes of real number sequences, $a = (a_1, a_2, a_3, \dots)$
  • These sequences must be Cauchy:
    • you give me an $\epsilon > 0$
    • I can give you can $N > 0$ such that
    • if you give me any $n > N $
    • we have $|a_n - a_m| < \epsilon$
  • Equivalence relation: $a \equiv b$ if for any $\epsilon > 0$ , there exist $N > 0$ with $|a_n - b_n|< \epsilon$ for all $n > N$.

Then we have to check these have the properties of the real numbers we like: $(a+b)+c = a + (b+c)$, least upper bound property, etc.

Oh, I tried writing the definition of Cauchy sequence as a "game". Any leads on that are welcome in the comments.

I had been reading about the divergent sequences 1 - 2 + 3 - 4 + ... and how to add them. The partial sums are 1, -1, 2, -2, 3, -3, ... which diverge. We can try averaging the partial sums get 1, 0, 2/3, 0, 3/5, 0, 5/7, 0,... which still doesn't converge

Maybe instead use the simpler example 1 -1 + 1 -1+... whose partial sums are 1,0,1,0,1,0 and we "claim" the sum should be "$\frac{1}{2}$". This is the Cesaro sum of the alternating sequence. What's iffy is that we can rearrange the sequence to -1+1-1+1-... which should tend to $-\frac{1}{2}$. But that rearrangement is "infinite" in some sense.

I would like to know about the collection infinite sequences of (say) rational numbers admitting Cesaro averages. We could say $a \equiv b$ if $a_n - b_n$ has Cesaro mean tending to $0$, I think this is another construction of the real numbers.

What if we just ask $a_n - b_n$ be Cauchy? What is the set of equivalence classes of rational numbers with Cesaro averages modulo the Cauchy sequences. It's probably not even Hausdorff.

Loosely related, there is a construction of the reals due to A'Campo of real numbers as "slopes" or "almost homomorphisms".

  • A "slope" is a sequence of number $ \lambda: \mathbb{Z} \to \mathbb{Z}$ with $\{ \lambda(m)+\lambda(n)-\lambda(m+n): m, n \in \mathbb{Z} \}$ a finite set.
  • Two slopes are "equivalent" if $\{\lambda(n) - \lambda(n') : n \in \mathbb{Z} \}$ is a finite set.
  • The "positive" slopes have finite $\{ n : \lambda(n) \leq 0\}$.
  • ${\bf 1}$ is the identity map ${\bf 1} : n \mapsto n$.

The sum is $\lambda(n) + \lambda'(n)$ and the product is $\lambda(\lambda'(n))$ with $n \in \mathbb{Z}$.

This idea is connected with rotation number of oriented homomorphisms of the circle.

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Thanks for sharing this "slopes" construction, it's interesting. – Zsbán Ambrus Jan 19 '12 at 14:49
The construction is originally due to Schanuel, not A’Campo. See for some historical remarks. – Emil Jeřábek Jan 19 '12 at 17:03
They were independent... and the emphases are different. Street/Schnauel are concerned with $\mathbb{R}$ as a logical construction. With A'Campo it seems to come out of geometry & topology. In any case, the integer sequence $\{n \mapsto [αn]\}$ maps to the real number $\alpha \in \mathbb{R}$. – john mangual Jan 19 '12 at 22:58

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