Proving Every Bounded Sequence in Real Coordinate Space Has a Convergent Subsequence

Understanding the behavior of sequences, particularly those that are bounded, is a fundamental concept in mathematical analysis. One of the key theorems in this area is the Bolzano-Weierstrass theorem. This theorem guarantees that every bounded sequence in the real coordinate space (mathbb{R}^n) contains a convergent subsequence. This article will walk you through the proof of this theorem, step by step.

Step 1: Define the Sequence

Consider a sequence (x_n) in the real coordinate space (mathbb{R}^n). We say that this sequence is bounded if there exists a positive constant (M) such that the norm of each term in the sequence is less than or equal to (M). Mathematically, this is expressed as:

Property 1: (|x_n| M) for all (n).

Step 2: Compactness in (mathbb{R}^n)

A crucial property of subsets in (mathbb{R}^n) is compactness. According to the Heine-Borel theorem, a subset of (mathbb{R}^n) is compact if and only if it is both closed and bounded. Since our sequence (x_n) is bounded, the set of points it occupies, ({x_n : n in mathbb{N}}), is also bounded. Therefore, this set is contained in a closed and bounded subset of (mathbb{R}^n), which is compact.

Step 3: Extracting a Subsequence

Given that the set of points ({x_n : n in mathbb{N}}) is compact, we can use the property of compact sets, which states that every sequence contained within a compact set has a convergent subsequence. Thus, we can extract a subsequence (x_{n_k}) from the original sequence (x_n) such that this subsequence converges to some limit (L) in (mathbb{R}^n). Mathematically, this is written as:

Property 2: (x_{n_k} rightarrow L) as (k rightarrow infty).

Conclusion

We have shown that every bounded sequence in (mathbb{R}^n) must contain a convergent subsequence. This proof relies on two key points:

The sequence is bounded. The set of points from the sequence is contained in a compact set. Any sequence in a compact set has a convergent subsequence.

Therefore, the Bolzano-Weierstrass theorem is a powerful tool in proving the existence of convergent subsequences within bounded sequences in the real coordinate space.

Summary of Key Points

Bounded Sequence: A sequence (x_n) in (mathbb{R}^n) is bounded if (|x_n| M). Compact Set: A set in (mathbb{R}^n) is compact if it is closed and bounded. Sequential Compactness: Any sequence in a compact set (mathbb{R}^n) has a convergent subsequence.

This proof demonstrates the importance of compactness in understanding the convergence behavior of bounded sequences.