Proving the Existence of a Convergent Subsequence for Bounded Sequences in mathbb{R} Using the Bolzano-Weierstrass Theorem

In this article, we will delve into the proof that every bounded sequence in the set of real numbers mathbb{R} has a convergent subsequence. This result is a fundamental theorem in real analysis and is known as the Bolzano-Weierstrass Theorem.

1. Definition of a Bounded Sequence

A sequence ((x_n)_{n1}^{infty}) in (mathbb{R}) is said to be bounded if there exists a real number (M geq 0) such that for all (n), [|x_n| leq M.] This means that all terms of the sequence lie within the interval ([-M, M]).

2. Application of the Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass theorem states that every bounded sequence in (mathbb{R}) has at least one convergent subsequence. This theorem is a cornerstone in understanding the behavior of bounded sequences in the context of real numbers.

3. Constructing the Proof

Bound the Sequence

Since the sequence ((x_n)_{n1}^{infty}) is bounded, we can denote the bounds as (m inf(x_n)) and (M sup(x_n)), where (m) and (M) are finite due to the boundedness of the sequence.

Choosing a Subsequence

The goal is to extract a subsequence that converges. We use the property of bounded sequences and the completeness of (mathbb{R}) to achieve this. The completeness of (mathbb{R}) ensures that there is a subsequence that can get arbitrarily close to a limit point.

Finding a Limit Point

Since the sequence is bounded, it has at least one limit point, defined as a point that can be approached by terms of the sequence. We will select a subsequence ((x_{n_k})) that converges to this limit point (L).

4. Formal Construction of the Subsequence

Approach to Limit:

Choose (L) as a limit point of the sequence. For any (epsilon > 0), there are infinitely many (n) such that [|x_n - L| Start with an initial index (n_1) such that [|x_{n_1} - L| Choose (n_2 > n_1) such that [|x_{n_2} - L| Continue this process to construct the subsequence ((x_{n_k})) where each term gets arbitrarily close to (L).

By the construction described above, we have shown that there exists a subsequence ((x_{n_k})) that converges to (L). Therefore, we conclude that every bounded sequence in (mathbb{R}) has a convergent subsequence.

5. Conclusion

The proof relies on the Bolzano-Weierstrass theorem, the properties of bounded sequences, and the completeness of (mathbb{R}) to demonstrate that a bounded sequence must have a convergent subsequence. This result is crucial for understanding the behavior of sequences in real analysis and has wide-ranging applications in mathematics.

Summary

In summary, the proof that every bounded sequence in (mathbb{R}) has a convergent subsequence is based on the Bolzano-Weierstrass theorem, properties of bounded sequences, and the completeness of (mathbb{R}).