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I have been told:

"The real numbers are defined to be the set of equivalence classes of pairs of rational sequences $(a_i,b_i)$, where (1) $\{a_i\}$ is increasing, (2) $\{b_i\}$ is decreasing, (3) for each $i=1,2,..., \hspace{2mm} b_i-a_i>0$, and (4) $\lim_{x \to \infty} (b_i - a_i)=0$."

I have then been asked to (1) prove the distributive law of multiplication for numbers, and (2) describe the representation of $\pi$ in this sense.

However, I am struggling with the definition as given to me, and thus I have no clue where to start for either of the other parts.

Any insight into any, or all, of these parts would be very much appreciated.

UserX
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    Oh my, that definition is horrible. It's some Frankenstein-like hybrid of Cantor's and Dedekind's definitions of the real numbers. – Asaf Karagila Nov 10 '13 at 23:57

1 Answers1

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Let $x\in\mathbb{R}$. $\exists (a_i,b_i)\in\mathbb{Q}^2$ that define $x$. Let $y\in\mathbb{R}$ be similarly defined by $(c_i,d_i)$, and $z\in\mathbb{R}$ by $(e_i,f_i)$. We can say $x=\lim_{i\to\infty} \frac{1}{2}(a_i+b_i)$. $$x(y+z)=\left(\lim_{i\to\infty}\frac{1}{2}(a_i+b_i)\right)\left(\lim_{i\to\infty}\frac{1}{2}(c_i+d_i)+\frac{1}{2}(e_i+f_i)\right)$$ Since all the numbers in the expression are rational, we can distribute: $$ x(y+z)=\lim_{i\to\infty}\frac{1}{4}(a_i+b_i)(c_i+d_i)+\frac{1}{4}(a_i+b_i)(e_i+f_i)=x\cdot y+x\cdot z $$

As for $\pi$, since $\pi=4\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$, you can use the odd terms as an increasing rational series and the even terms as a decreasing one that converges on $\pi$.

Tim Ratigan
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