Consider the set of all Cauchy sequences of rational numbers: concretely, the set X={(an)∞n=1:an∈Q,(∀ϵ∈Q>0)(∃N∈N)(∀n,m∈N>N)(|an−am|<ϵ)}
Define an equivalence relation on this set, by (an)∼(bn) if and only if, for every rational ϵ>0, there is a natural number N such that for all n∈N bigger than N, we have |an−bn|<ϵ.
This is an equivalence relation (exercise).
Show solution
- It is symmetric, because |an−bn|=|bn−an|.
- It is reflexive, because |an−an|=0 for every n, and this is <ϵ.
- It is transitive, because if |an−bn|<ϵ2 for sufficiently large n, and |bn−cn|<ϵ2 for sufficiently large n, then |an−bn|+|bn−cn|<ϵ2+ϵ2=ϵ for sufficiently large n; so by the triangle_inequality, |an−cn|<ϵ for sufficiently large n.
Write [an] for the equivalence class of (an)∞n=1. (This is a slight abuse of notation, omitting the brackets that indicate that an is actually a sequence rather than a rational number.)
The set of real numbers is the set of equivalence classes of X under this equivalence relation, endowed with the following totally ordered field structure:
- [an]+[bn]:=[an+bn]
- [an]×[bn]:=[an×bn]
- [an]≤[bn] if and only if [an]=[bn] or there is some N such that for all n>N, an≤bn.
This field structure is well-defined (proof).
Examples
- Any rational number r may be viewed as a real number, being the class [r] (formally, the equivalence class of the sequence (r,r,…)).
- The real number π is indeed a real number under this definition; it is represented by, for instance, (3,3.1,3.14,3.141,…). It is also represented as (100,3,3.1,3.14,…), along with many other possibilities.