Every Bounded Sequence Has A Convergent Subsequence

Every Bounded Sequence Has a Convergent Subsequence: A Proof and Explanation

This article delves into the fundamental theorem of real analysis: every bounded sequence has a convergent subsequence. Understanding this theorem is crucial for grasping concepts like compactness and the Bolzano-Weierstrass theorem. We'll explore the proof and its implications, making it accessible to both beginners and those looking for a refresher.

This theorem essentially states that within any sequence whose terms remain within a specific range (bounded), we can always find a subsequence that approaches a limit. This seemingly simple statement has profound consequences in mathematical analysis.

Understanding the Concepts

Before diving into the proof, let's clarify the key terms:

• Sequence: An ordered list of numbers (terms). We usually denote a sequence as {a_n}, where a_n represents the nth term.
• Bounded Sequence: A sequence is bounded if there exists a real number M such that |a_n| ≤ M for all n. In simpler terms, all terms of the sequence lie within a specific interval.
• Subsequence: A subsequence is a sequence derived from another sequence by selecting a subset of its terms, while maintaining the original order. For example, if we have the sequence {1, 2, 3, 4, 5}, {1, 3, 5} is a subsequence.
• Convergent Sequence: A sequence converges to a limit L if, for any small positive number ε, there exists an integer N such that |a_n - L| < ε for all n > N. In essence, the terms of the sequence get arbitrarily close to L as n approaches infinity.

The Proof: A Step-by-Step Approach

We will prove this theorem using the Bolzano-Weierstrass Theorem which states that every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence. For simplicity, we'll focus on the case in $\mathbb{R}$ (real numbers).

1. Boundedness: We start with a bounded sequence {a_n}. This means there exists a real number M such that |a_n| ≤ M for all n.

2. Nested Intervals: We can divide the interval [-M, M] into two equal subintervals. At least one of these subintervals contains infinitely many terms of the sequence. Let's call this interval I₁.

3. Iterative Process: We repeat the process, dividing I₁ into two equal subintervals. Again, at least one of these subintervals contains infinitely many terms of the sequence. Let's call this interval I₂. We continue this process, creating a nested sequence of intervals I₁ ⊇ I₂ ⊇ I₃ ⊇ ...

4. Nested Interval Theorem: The nested interval theorem states that if we have a nested sequence of closed intervals whose lengths approach zero, then there exists exactly one point common to all intervals.

5. Subsequence Construction: From the infinitely many terms in I₁, we choose a_n₁. From the infinitely many terms in I₂, we choose a_n₂ such that n₂ > n₁. We continue this process, selecting a_{n_k} from I_k such that n_k > n_k-1. This creates a subsequence {a_{n_k}}.

6. Convergence: The length of the intervals I_k approaches zero as k approaches infinity. Therefore, the subsequence {a_{n_k}} converges to the unique point common to all intervals I_k.

Implications and Applications

The theorem that every bounded sequence has a convergent subsequence is a cornerstone of real analysis. It plays a vital role in proving other important theorems and has numerous applications in:

• Calculus: Understanding limits and convergence of sequences and series.
• Numerical Analysis: Developing algorithms for finding approximations of solutions to equations.
• Optimization: Finding optimal solutions to problems involving sequences of approximations.
• Probability and Statistics: Analyzing the convergence of random variables.

Conclusion

The proof that every bounded sequence has a convergent subsequence, while requiring careful consideration, is a powerful tool in mathematics. Its elegance and importance lie in its ability to guarantee the existence of a convergent subsequence within any bounded sequence, providing a solid foundation for many advanced mathematical concepts. Understanding this theorem enhances the comprehension of limits, convergence, and the broader landscape of real analysis.