The reals (constructed as classes of Cauchy sequences of rationals) form a field

The real numbers, when constructed as equivalence classes of Cauchy sequences of rationals, form a totally ordered field, with the inherited field structure given by

• $$[a_n] + [b_n] = [a_n+b_n]$$

• $$[a_n] \times [b_n] = [a_n \times b_n]$$

• $$[a_n] \leq [b_n]$$ if and only if either $$[a_n] = [b_n]$$ or for sufficiently large $$n$$, $$a_n \leq b_n$$.

Proof

Firstly, we need to show that those operations are even well-defined: that is, if we pick two different representatives $$(x_n)_{n=1}^{\infty}$$ and $$(y_n)_{n=1}^{\infty}$$ of the same equivalence class $$[x_n] = [y_n]$$, we don’t somehow get different answers.

Well-definedness of $$+$$

We wish to show that $$[x_n]+[a_n] = [y_n] + [b_n]$$ whenever $$[x_n] = [y_n]$$ and $$[a_n] = [b_n]$$; this is an exercise.

Since $$[x_n] = [y_n]$$, it must be the case that both $$(x_n)$$ and $$(y_n)$$ are Cauchy sequences such that $$x_n - y_n \to 0$$ as $$n \to \infty$$. Similarly, $$a_n - b_n \to 0$$ as $$n \to \infty$$.

We require $$[x_n+a_n] = [y_n+b_n]$$; that is, we require $$x_n+a_n - y_n-b_n \to 0$$ as $$n \to \infty$$.

But this is true: if we fix rational $$\epsilon > 0$$, we can find $$N_1$$ such that for all $$n > N_1$$, we have $$|x_n - y_n| < \frac{\epsilon}{2}$$; and we can find $$N_2$$ such that for all $$n > N_2$$, we have $$|a_n - b_n| < \frac{\epsilon}{2}$$. Letting $$N$$ be the maximum of the two $$N_1, N_2$$, we have that for all $$n > N$$, $$|x_n + a_n - y_n - b_n| \leq |x_n - y_n| + |a_n - b_n|$$ by the triangle inequality, and hence $$\leq \epsilon$$. <div><div>

Well-definedness of $$\times$$

We wish to show that $$[x_n] \times [a_n] = [y_n] \times [b_n]$$ whenever $$[x_n] = [y_n]$$ and $$[a_n] = [b_n]$$; this is also an exercise.

We require $$[x_n a_n] = [y_n b_n]$$; that is, $$x_n a_n - y_n b_n \to 0$$ as $$n \to \infty$$.

Ring structure

The multiplicative identity is $$[1]$$ (formally, the equivalence class of the sequence $$(1,1, \dots)$$). Indeed, $$[a_n] \times [1] = [a_n \times 1] = [a_n]$$.

$$\times$$ is closed, because the product of two Cauchy sequences is a Cauchy sequence (exercise).

If $$(a_n)$$ and $$(b_n)$$ are Cauchy sequences, then let $$\epsilon > 0$$. We wish to show that there is $$N$$ such that for all $$n, m > N$$, we have $$|a_n b_n - a_m b_m| < \epsilon$$.

But $$|a_n b_n - a_m b_m| = |a_n (b_n - b_m) + b_m (a_n - a_m)| \leq |b_m| |a_n - a_m| + |a_n| |b_n - b_m|$$$by the triangle inequality. Cauchy sequences are bounded, so there is $$B$$ such that $$|a_n|$$ and $$|b_m|$$ are both less than $$B$$ for all $$n$$ and $$m$$. So picking $$N$$ so that $$|a_n - a_m| < \frac{\epsilon}{2B}$$ and $$|b_n - b_m| < \frac{\epsilon}{2B}$$ for all $$n, m > N$$, the result follows. <div><div> $$\times$$ is clearly commutative: $$[a_n] \times [b_n] = [a_n \times b_n] = [b_n \times a_n] = [b_n] \times [a_n]$$. $$\times$$ is associative: $$[a_n] \times ([b_n] \times [c_n]) = [a_n] \times [b_n \times c_n] = [a_n \times b_n \times c_n] = [a_n \times b_n] \times [c_n] = ([a_n] \times [b_n]) \times [c_n]$$$

$$\times$$ distributes over $$+$$: we need to show that $$[x_n] \times ([a_n]+[b_n]) = [x_n] \times [a_n] + [x_n] \times [b_n]$$. But this is true: $$[x_n] \times ([a_n]+[b_n]) = [x_n] \times [a_n+b_n] = [x_n \times (a_n+b_n)] = [x_n \times a_n + x_n \times b_n] = [x_n \times a_n] + [x_n \times b_n] = [x_n] \times [a_n] + [x_n] \times [b_n]$$\$

Field structure

To get from a ring to a field, it is necessary and sufficient to find a multiplicative inverse for any $$[a_n]$$ not equal to $$[0]$$.

Since $$[a_n] \not = 0$$, there is some $$N$$ such that for all $$n > N$$, $$a_n \not = 0$$. Then defining the sequence $$b_i = 1$$ for $$i \leq N$$, and $$b_i = \frac{1}{a_i}$$ for $$i > N$$, we obtain a sequence which induces an element $$[b_n]$$ of $$\mathbb{R}$$; and it is easy to check that $$[a_n] [b_n] = [1]$$.

$$[a_n] [b_n] = [a_n b_n]$$; but the sequence $$(a_n b_n)$$ is $$1$$ for all $$n > N$$, and so it lies in the same equivalence class as the sequence $$(1, 1, \dots)$$.

Ordering on the field

We need to show that:

• if $$[a_n] \leq [b_n]$$, then for every $$[c_n]$$ we have $$[a_n] + [c_n] \leq [b_n] + [c_n]$$;

• if $$[0] \leq [a_n]$$ and $$[0] \leq [b_n]$$, then $$[0] \leq [a_n] \times[b_n]$$.

We may assume that the inequalities are strict, because if equality holds in the assumption then everything is obvious.

If $$[a_n] = [b_n]$$, then for every $$[c_n]$$ we have $$[a_n] + [c_n] = [b_n] + [c_n]$$ by well-definedness of addition. Therefore $$[a_n] + [c_n] \leq [b_n] + [c_n]$$.

If $$[0] = [a_n]$$ and $$[0] \leq [b_n]$$, then $$[0] = [0] \times [b_n] = [a_n] \times [b_n]$$, so it is certainly true that $$[0] \leq [a_n] \times [b_n]$$. <div><div>

For the former: suppose $$[a_n] < [b_n]$$, and let $$[c_n]$$ be an arbitrary equivalence class. Then $$[a_n] + [c_n] = [a_n+c_n]$$; $$[b_n] + [c_n] = [b_n+c_n]$$; but we have $$a_n + c_n \leq b_n + c_n$$ for all sufficiently large $$n$$, because $$a_n \leq b_n$$ for sufficiently large $$n$$. Therefore $$[a_n] + [c_n] \leq [b_n] + [c_n]$$, as required.

For the latter: suppose $$[0] < [a_n]$$ and $$[0] < [b_n]$$. Then for sufficiently large $$n$$, we have both $$a_n$$ and $$b_n$$ are positive; so for sufficiently large $$n$$, we have $$a_n b_n \geq 0$$. But that is just saying that $$[0] \leq [a_n] \times [b_n]$$, as required.

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