>>11042172I understand that my post was probably the first time a lot of people here saw what real numbers actually were. I will take this opportunity to teach you some more. We will continue with the Cauchy definition of real numbers. How do we define the field operations?
Simple. From now on, we will refer to real numbers by arbitrary elements in their equivalence classes. This means you have to prove that the definitions are well defined independent of the choice of element in the equivalence class, but that can be done without too much effort.
The most basic operation is addition. We define (a_n)+(b_n) be (a_n + b_n) (essentially sum each entry in the sequence. so (1,1,1,1,1,1,...) + (1, 1/2, 1/4, 1/8,...) will be (1, 1.5, 1.25, ....). We need to prove that this defines a unique real number independent of choice of sequences (a_n) and (b_n) in their respective equivalence classes. This is easy: assume (a_n) ~ (a'_n) and (b_n) ~ (b'_n). We need to show that (a_n + b_n)~(a'_n + b'_n), or equivalently that |a_n + b_n - a'_n + b'_n | -> 0 as n-> infinity. But |a_n + b_n - a'_n + b'_n | = | (a_n - a'_n) + (b_n - b'_n)| <= |a_n - a'_n| + |b_n - b'_n| (here we used the triangle inequality |a+b| <= |a| + |b|). Hence since |a_n - a'_n| tends to 0, as does |b_n - b'_n| , their sum does too, hence (a_n + b_n)~(a'_n + b'_n) and addition of real numbers is well defined. This is actually not the complete proof yet: we need to establish that addition actually defines a real number by showing that (a_n + b_n) is also a Cauchy sequence, but that proof is similar and left as an exercise to the reader :)
Now that you've seen how addition is defined, try and yourself define: addition, multiplication and division. Not just tell us how to do it, but show that your definition actually works, as I did.
