Proving Convergence of Sequences: A Detailed Analysis

Proving Convergence of Sequences: A Detailed Analysis

Understanding the concept of convergence in sequences is fundamental in mathematical analysis. This article delves into the process of proving that a sequence is convergent, using the sequence an 1/n as an example, and demonstrating how to show that it satisfies the Cauchy criterion.

Introduction to Convergent Sequences

A sequence of real numbers {an} is said to be convergent if it approaches a finite limit as n tends to infinity. In other words, there exists a number L such that an → L as n → ∞.

Proving Convergence Using the Cauchy Criterion

Another criterion for determining the convergence of a sequence is the Cauchy criterion. A sequence {an} is called a Cauchy sequence if for every positive ε, there exists an integer N such that for all n, m ≥ N, we have |an - am| . If a Cauchy sequence of real numbers is also a bounded sequence, then it is convergent. This is a powerful tool to prove the convergence of a sequence without explicitly finding the limit.

An Example: Proving Convergence of an 1/n

Let's consider the sequence an 1/n. We will prove that this sequence is convergent by showing that it is a Cauchy sequence. The goal is to find such an N for any given ε > 0 such that |1/n - 1/m| for all n, m ≥ N.

Step 1: Setting Up the Inequality

First, we need to manipulate the expression |1/n - 1/m|. Considering two cases: n m, and m ≤ n.

For n ≤ m, we have:

|1/n - 1/m| |1/n - 1/m| |(m - n) / (nm)|

Given the nature of the inequality, we can further simplify this to:

|1/n - 1/m| |1/n - 1/m| ≤ 1/n 1/m

Step 2: Finding an Appropriate N

Now, let's consider the inequality 1/n 1/m . We want to find an N such that this inequality holds for all n, m ≥ N. For simplicity, let's set N ≥ 2/ε. Then, for any n, m ≥ N, we have:

1/n 1/m ≤ 1/N 1/N 2/N ≤ 2/(2/ε) ε

This holds true as long as N ≥ 2/ε. Therefore, we have found an N for any arbitrary ε > 0 such that |1/n - 1/m| for all n, m ≥ N.

Conclusion: Proving Convergence

We have shown that the sequence an 1/n is a Cauchy sequence. Since every Cauchy sequence in ? (the set of real numbers) that is also bounded is convergent, we can conclude that the sequence an 1/n is indeed convergent.

In summary, proving the convergence of a sequence using the Cauchy criterion involves finding an appropriate N for any given ε > 0 such that the sequence satisfies the inequality |an - am| for all n, m ≥ N. For the sequence an 1/n, we found such an N as N ≥ 2/ε, demonstrating its convergence.

Related Keywords

Keyword 1: Convergent sequence

A sequence that approaches a finite limit as n tends to infinity.

Keyword 2: Cauchy sequence

A sequence where the terms become arbitrarily close to each other as the sequence progresses.

Keyword 3: Mathematical proof

A rigorous method of demonstrating the truth of a mathematical statement using logical deductions from known facts or axioms.