18.099: Problem Set 6

Due: Tuesday, November 4.

The purpose this time is to write a small paper about Cantor’s construction of the real numbers. In short, this is the construction of the real numbers as the completion of the rational numbers with respect to the (usual) euclidean metric.

We consider sequences {a_{n}} of rational numbers. We say that the sequence {a_{n}} converges to a ∈ ℚ if for all ϵ > 0 (ϵ ∈ ℚ), there exists a positive integer N such that for all n ≥ N,

|a − a_{n}| < ϵ.

Similarly, we say that {a_{n}} is a Cauchy sequence if for all ϵ > 0 (ϵ ∈ ℚ), there exists a positive integer N such that for all m, n ≥ N,

|a_{m} − a_{n}| < ϵ.

These are the definitions except that we only consider ϵ > 0 that are rational numbers. (We are constructing the real numbers.)

We say that two Cauchy sequences of rational numbers {a_{n}} and {b_{n}} are equivalent and write {a_{n}} ∼ {b_{n}} if the sequence {a_{n}−b_{n}} converges to zero. Show that this relation is an equivalence relation, that is, the following three conditions are satisfied.

Symmetry: {a_{n}} ∼ {a_{n}}.

Reflexivity: {a_{n}} ∼ {b_{n}} implies {b_{n}} ∼ {a_{n}}.

Transitivity: {a_{n}} ∼ {b_{n}} and {b_{n}} ∼ {c_{n}} implies {a_{n}} ∼ {c_{n}}.

In general, if we are given an equivalence relation on a set X, then a subset A ⊂ X is called an equivalence class if the following two conditions are satisfied.

(i) a, a′ ∈ A implies a ∼ a′.

(ii) a ∈ A, x ∈ X and a ∼ x implies x ∈ A.

Show that every element x ∈ X belongs to a unique equivalence class.

Consider again the set of Cauchy sequences of rational numbers with the equivalence relation defined earlier. Following Cantor, we define a real number to an equivalence class of Cauchy sequences of rational numbers. So the set ℝ is the set of equivalence classes of Cauchy sequences of rational numbers. The addition and multiplication on the set of real numbers is given by term-wise adding and multiplying Cauchy sequences of rational numbers. Show that the addition and multiplication is well-defined. Show that ℝ is a field and identify ℚ with a subfield of ℝ.

We next define the order on the set or real numbers. A Cauchy sequence of rational numbers {a_{n}} is called positive if there exists a rational number ϵ > 0 and a positive integer N such that for all n ≥ N, a_{n} > ϵ. Show that if {a_{n}} ∼ {b_{n}} and if {a_{n}} is positive, then {b_{n}} is positive. We then define x ∈ ℝ to be positive if {a_{n}} ∈ x is positive. Show that for every x ∈ ℝ exactly one of the following three statements is true.

(i) x = 0.

(ii) x is positive.

(iii) −x is positive.

Show that with this ordering ℝ is an ordered field.

Let {a_{n}} ∈ x ∈ ℝ. Show that if we consider {a_{n}} as a Cauchy sequence of real numbers, then {a_{n}} converges to x. Show that ℝ is complete. Show that ℝ has the least upper bound property.