The Real Number System

The real number system, $\mathbb{R}$, is a ring, $R,$ definined on the quotient set of all rational Cauchy sequences such that the operations of addition and multiplication are defined as follows:

$$[a_n] + [b_n] = [a_n + b_n] \\ [a_n][b_n] = [a_n b_n] $$

The Cauchy sequences are in brackets to denote the equivalence classes. For example, 1/2 and 3/6 would be in the same equivalence class. The quotient set of $a_n$ would be the set of all equivalence classes whose limit is $a$.

From the above definition it follows that the real number system is a field because it can easily be shown $\mathbb{R}$ is a commutative ring with unit. For example, the property of commutativity follows from the way the real numbers were defined above. The same goes for commutative multiplication and associativity.

The element 0 is an element of $\mathbb{R}$ since $a_n$ can be defined as the sequence consisting of all zeroes. It can also be shown if $a_n \neq 0,$ then $a_n$ has a multiplicative inverse. We can make $\mathbb{R}$ an ordered field by defining a set of positive numbers which are positive Cauchy sequences. The positive Cauchy sequences have the same properties as defined in the positive cone earlier.

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