Question

Prove: A bounded sequence in \(\mathbb{R}^{n}\) has a convergent subsequence.

Short answer

Question: Prove that a bounded sequence in \(\mathbb{R}^n\) has a convergent subsequence. Answer: By using the Bolzano-Weierstrass theorem, which states that every bounded sequence in \(\mathbb{R}^n\) contains a convergent subsequence, we can extract convergent subsequences for each of the sequence's components. This yields a convergent subsequence of the original bounded sequence in \(\mathbb{R}^n\).

Step-by-step solution

Definition of a bounded sequence in \(\mathbb{R}^n\)

A sequence \((x_k)\) in \(\mathbb{R}^n\) is called bounded if there exists a positive number \(M\) such that \(\|x_k\| \leq M\) for all \(k \in \mathbb{N}\). In other words, all the terms of the sequence lie inside a closed ball of radius \(M\) centered at the origin.

Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass theorem states that every bounded sequence in \(\mathbb{R}^n\) contains a convergent subsequence. In other words, given any bounded sequence \((x_k)\) in \(\mathbb{R}^n\), there exists a subsequence \((x_{k_i})\) and a limit point \(x \in \mathbb{R}^n\) such that for every \(\epsilon > 0\), there is a positive integer \(N\) such that \(\|x_{k_i} - x\| < \epsilon\) for all \(i > N\).

Extracting a convergent subsequence from the bounded sequence

Since \((x_k)\) is a bounded sequence in \(\mathbb{R}^n\), according to the Bolzano-Weierstrass theorem, it contains a convergent subsequence \((x_{k_i})\). To demonstrate this, we can first consider the sequence of the first components of all \(x_k\), let's call it \((x_{k,1})\). Since \((x_k)\) is bounded, so are \((x_{k,1})\). Thus, by the theorem, \((x_{k,1})\) has a convergent subsequence \((x_{k'_i,1})\). Now consider the sequence of the second components of \(x_{k'}\), let's call it \((x_{k'_i,2})\). Similar to the argument above, this sequence is also bounded, and thus has a convergent subsequence \((x_{k''_i,2})\). We continue this process for all \(n\) components to construct convergent subsequences of each component's sequence. Define \(k_i = k_{i,i}\), then \((x_{k_i})\) is the desired convergent subsequence of \((x_k)\).

Conclusion

A bounded sequence \((x_k)\) in \(\mathbb{R}^n\) indeed has a convergent subsequence \((x_{k_i})\), as shown via the extraction of convergent subsequences for each of its components using the Bolzano-Weierstrass theorem.

Key concepts

Bounded Sequence in \(\mathbb{R}^n\)

When we talk about sequences in the context of real analysis, particularly in higher dimensions like \(\mathbb{R}^n\), understanding the concept of 'boundedness' is crucial. A sequence \((x_k)\) consisting of vectors in \(\mathbb{R}^n\) is called a bounded sequence if all elements of the sequence are contained within a certain 'distance' from the origin. This distance is represented by some positive number \(M\). Thus, mathematically speaking, a sequence is bounded if \(\|x_k\| \leq M\) for all \(k\) in the sequence, utilizing the norm \(\|\cdot\|\) to measure 'distance' in \(\mathbb{R}^n\).

This concept ensures that the vectors do not 'escape' to infinity in any direction within the n-dimensional space. The visual interpretation is that every vector in the sequence lies inside or on the surface of a multidimensional sphere (an n-ball) centered at the origin with a radius \(M\).

Bolzano-Weierstrass Theorem

One of the cornerstones of real analysis is the Bolzano-Weierstrass theorem. It's a fundamental result with profound implications, one of which is particularly relevant for our exercise. The theorem states that in any bounded sequence in \(\mathbb{R}^n\), there exists at least one convergent subsequence.

This theorem helps us to guarantee that no matter how 'erratic' a sequence may seem, as long as it stays within a certain bound, there will be some order to be found in the form of a convergent subsequence. It's like finding a pattern or a repeatable trend within what appears to be complete chaos. For students diving into real analysis, an intuitive grasp of the Bolzano-Weierstrass theorem is essential as it frequently is the key step in many proofs and exercises involving sequences and their properties.

Real Analysis

Real analysis is a significant branch of mathematics that deals with the behavior and properties of real numbers, sequences, series, and real-valued functions. It lays down the rigorous framework for limits, continuity, differentiation, and integration which are the foundations of calculus. All of these concepts are woven together by intricate theorems and proofs. One such theorem, the Bolzano-Weierstrass theorem, plays a vital role in understanding elements of real analysis like convergence, sequences, and more. The beauty of real analysis is in its ability to formulate and solve problems using precise definitions and logical reasoning, equipping students to tackle more complex mathematical challenges.

Sequence Convergence

The concept of sequence convergence is the heart of many real analysis discussions. A sequence \((x_k)\) in \(\mathbb{R}^n\) is said to converge to a limit \(x\) if its elements get arbitrarily close to \(x\) as you move further along the sequence. Formally, we say that \((x_k)\) converges to \(x\) as \(k\) approaches infinity if, given any tiny positive distance \(\epsilon\), there exists a stage in the sequence \(N\) after which all elements of the sequence are within \(\epsilon\) from the limit \(x\).

The convergence of a sequence shows how a series of potentially complex and multi-dimensional values can settle into a stable pattern, represented by a single point in \(\mathbb{R}^n\). Understanding convergence is central to not just sequences, but the entire scope of calculus and analysis, ultimately leading to better comprehension of the behavior of various mathematical and physical systems.