Proving the Completeness of Real Numbers Through Cauchy Sequences

In mathematics, especially within the domain of real analysis, the completeness property of the real numbers is a fundamental concept. This proof focuses on demonstrating that the real numbers (mathbb{R}) are complete by showing that every Cauchy sequence of real numbers converges to a limit, which is also a real number. This article provides a detailed exploration of the proof, including the definitions, the proof outline, and the reasoning behind each step.

Definitions

Cauchy Sequence

A sequence (x_n) in (mathbb{R}) is called a Cauchy sequence if for every (epsilon > 0), there exists an integer (N) such that for all (m, n geq N), we have (|x_m - x_n|

Completeness

A metric space is said to be complete if every Cauchy sequence in that space converges to a limit within the same space. This property is crucial for understanding the structure and behavior of the space, particularly in terms of continuity and convergence.

Proof Outline

Given a Cauchy Sequence

Let (x_n) be a Cauchy sequence in (mathbb{R}).

Boundedness

Since (x_n) is a Cauchy sequence, it is also bounded. To prove this, we proceed as follows:

By the definition of a Cauchy sequence, for any (epsilon 1), there exists an integer (N) such that for all (m, n geq N), we have (|x_m - x_n| Choose (x_N) as a reference point. For (n geq N), we have:

(|x_n - x_N| Thus, the sequence is bounded between (x_N - 1) and (x_N 1). For the terms before (N), since they are finite in number, we can find a maximum and minimum ensuring the entire sequence is bounded.

Existence of a Limit

Since the sequence (x_n) is bounded, by the Bolzano-Weierstrass theorem, it has a convergent subsequence. Let's denote this subsequence by (x_{n_k}) which converges to some limit (L).

Convergence of the Entire Sequence

We need to show that the entire sequence (x_n) converges to (L). For any (epsilon > 0), there exists (K) such that for all (k geq K), we have (|x_{n_k} - L| Since (x_n) is a Cauchy sequence, there exists (M) such that for all (m, n geq M), we have (|x_m - x_n| Choose (N max(M, n_K)). For (n geq N), we can distinguish two cases: Case 1: If (n n_k) for some (k geq K), then: (|x_n - L| Case 2: If (n) is not in the subsequence, we can find some (n_j) such that (j geq K) and (m geq N). Using the triangle inequality: (|x_n - L| leq |x_n - x_{n_k}| |x_{n_k} - L|

Since we can make (|x_n - L|

Summary

The completeness of (mathbb{R}) is established by demonstrating that every Cauchy sequence converges to a limit within (mathbb{R}) through boundedness and the application of the Bolzano-Weierstrass theorem.