Every Bounded Sequence Has A Convergent Subsequence

Every Bounded Sequence Has a Convergent Subsequence: A Deep Dive into the Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass Theorem is a cornerstone of real analysis, stating that every bounded sequence in $\mathbb{R}^n$ (or more generally, in any complete metric space) has a convergent subsequence. This seemingly simple statement has profound implications across various mathematical fields, from calculus to functional analysis. This article will delve into a detailed proof of this theorem, exploring its nuances and demonstrating its power through examples and applications. We will also touch upon its generalizations and related concepts.

Understanding the Concepts

Before diving into the proof, let's solidify our understanding of the key concepts:

1. Sequences:

A sequence is an ordered list of numbers (or elements from a set). We often denote a sequence as ${x_n}_{n=1}^{\infty}$ or simply ${x_n}$, where $x_n$ represents the $n$-th term of the sequence.

2. Bounded Sequences:

A sequence ${x_n}$ is said to be bounded if there exists a real number $M > 0$ such that $|x_n| \leq M$ for all $n$. In essence, all terms of the sequence lie within a finite interval.

3. Convergent Sequences:

A sequence ${x_n}$ is said to be convergent if there exists a real number $L$ (called the limit) such that for every $\epsilon > 0$, there exists an integer $N$ such that for all $n > N$, $|x_n - L| < \epsilon$. Intuitively, this means that the terms of the sequence get arbitrarily close to $L$ as $n$ approaches infinity.

4. Subsequences:

A subsequence of a sequence ${x_n}$ is a sequence formed by selecting some (possibly infinitely many) terms from the original sequence, maintaining their original order. For example, if ${x_n} = {1, 2, 3, 4, 5, ...}$, then ${x_{2n}} = {2, 4, 6, 8, ...}$ is a subsequence. Formally, a subsequence is denoted as ${x_{n_k}}_{k=1}^{\infty}$ where ${n_k}$ is a strictly increasing sequence of natural numbers.

Proving the Bolzano-Weierstrass Theorem

We will prove the theorem for the case of bounded sequences in $\mathbb{R}$. The extension to $\mathbb{R}^n$ is straightforward using a similar approach applied to each coordinate.

Theorem: Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.

Proof:

Let ${x_n}$ be a bounded sequence in $\mathbb{R}$. By definition, there exists $M > 0$ such that $|x_n| \leq M$ for all $n$. This means that the sequence is contained within the interval $[-M, M]$.

We will employ the method of bisection.

1. Divide and Conquer: Divide the interval $[-M, M]$ into two equal subintervals: $[-M, 0]$ and $[0, M]$. At least one of these subintervals must contain infinitely many terms of the sequence ${x_n}$. Let's call this interval $I_1$.

2. Iterative Bisection: Now bisect $I_1$ into two equal subintervals. Again, at least one of these subintervals contains infinitely many terms of the sequence. Call this interval $I_2$.

3. Continue the Process: We continue this process iteratively. At the $k$-th step, we have an interval $I_k$ of length $\frac{2M}{2^k}$ containing infinitely many terms of the sequence.

4. Nested Intervals: The sequence of intervals ${I_k}$ forms a nested sequence of closed intervals, i.e., $I_1 \supseteq I_2 \supseteq I_3 \supseteq ...$ The length of these intervals approaches 0 as $k \to \infty$.

5. The Nested Interval Theorem: By the Nested Interval Theorem (a fundamental result in real analysis), there exists a unique point $L$ that is contained in all $I_k$.

6. Constructing the Subsequence: Now, we construct a subsequence ${x_{n_k}}$ as follows:

   • Choose $x_{n_1}$ to be any term of the sequence in $I_1$.
   • Choose $x_{n_2}$ to be any term of the sequence in $I_2$ such that $n_2 > n_1$.
   • Continue this process. At the $k$-th step, choose $x_{n_k}$ to be any term of the sequence in $I_k$ such that $n_k > n_{k-1}$. This is always possible because each $I_k$ contains infinitely many terms.

7. Convergence of the Subsequence: Since $x_{n_k} \in I_k$ for all $k$, and the length of $I_k$ approaches 0, it follows that the subsequence ${x_{n_k}}$ converges to $L$.

Therefore, every bounded sequence in $\mathbb{R}$ has a convergent subsequence. This completes the proof.

Implications and Applications

The Bolzano-Weierstrass Theorem is a powerful tool with widespread applications:

1. Existence of Limits:

The theorem guarantees the existence of convergent subsequences, even when the original sequence may not converge. This is crucial in establishing the existence of limits in various contexts.

2. Calculus:

The theorem underpins many results in calculus, particularly those related to limits and continuity. For example, it's used in proving the Extreme Value Theorem, which states that a continuous function on a closed interval attains its maximum and minimum values.

3. Optimization Problems:

In optimization problems, the theorem can be used to show the existence of optimal solutions. If a sequence of approximate solutions is bounded, the theorem ensures the existence of a convergent subsequence, and the limit of this subsequence may be an optimal solution.

4. Functional Analysis:

The theorem is generalized to complete metric spaces in functional analysis, where it plays a vital role in proving many fundamental results in operator theory and other areas.

5. Numerical Analysis:

In numerical analysis, the theorem helps justify the convergence of iterative methods. If the sequence of iterates is bounded, the theorem guarantees the existence of a convergent subsequence, providing a basis for convergence analysis.

Generalizations and Related Concepts

The Bolzano-Weierstrass Theorem is a special case of more general results:

1. Complete Metric Spaces:

The theorem generalizes to complete metric spaces. In a complete metric space, every bounded sequence has a convergent subsequence. This generalization is extremely important in functional analysis.

2. Compactness:

The theorem is closely related to the concept of compactness. In a metric space, compactness is equivalent to sequential compactness, meaning that every sequence in a compact set has a convergent subsequence.

Conclusion

The Bolzano-Weierstrass Theorem is a fundamental result in real analysis with far-reaching consequences. Its elegant proof, based on the method of bisection and the nested interval theorem, showcases the power of fundamental concepts in constructing a powerful result. The theorem's generalizations and connections to other important concepts in analysis highlight its centrality to the study of infinite sequences and their limits. Its applications extend across various mathematical disciplines, solidifying its place as a cornerstone of modern mathematical analysis. Understanding and appreciating this theorem are crucial for anyone pursuing advanced studies in mathematics or related fields. The theorem's simplicity belies its profound implications, making it a beautiful and important result worthy of deep study.