Understanding Bounded Sequences and Their Properties

In mathematics, specifically in real analysis, a sequence is considered bounded sequence if there exists a real number \(M\) such that the absolute value of every term in the sequence is less than or equal to \(M\). This means the sequence is contained within a finite interval on the number line.

Every Bounded Sequence Possesses a Convergent Subsequence

The question asks what every bounded sequence has. Let's look at the options.

Option 1: A divergent subsequence. A bounded sequence might have a divergent subsequence (for example, the sequence \((-1)^n\) is bounded and has subsequences like \((-1)^{2k} = 1, 1, 1, ...\) which converges, and \((-1)^{2k+1} = -1, -1, -1, ...\) which also converges. Consider a sequence like \(1, -1, 2, -2, 3, -3, ...\). This sequence is not bounded. Consider a bounded sequence like \(1, 0, 1, 0, 1, 0, ...\). It has convergent subsequences \(1, 1, 1, ...\) and \(0, 0, 0, ...\). A bounded sequence doesn't necessarily have a divergent subsequence. For example, a sequence that converges itself is also bounded and all its subsequences converge.

Option 3: A divergent sequence. A bounded sequence does not necessarily have to be a divergent sequence. For instance, the sequence \(a_n = 1/n\) for \(n \ge 1\) is a bounded sequence (bounded between 0 and 1) and it converges to 0.

Option 2: A convergent subsequence. This is the correct property that every bounded sequence in \(\mathbb{R}\) possesses.

The Bolzano-Weierstrass Theorem

The property that every bounded sequence in \(\mathbb{R}\) has a convergent subsequence is a fundamental result in real analysis. It is formally stated by the Bolzano-Weierstrass theorem.

The Bolzano-Weierstrass theorem states: Every bounded sequence of real numbers has at least one convergent subsequence.

This theorem is a cornerstone of real analysis and highlights a key property of the real number system, specifically its completeness.

Let's consider the sequence \((-1)^n\). This is a bounded sequence as \(-1 \le (-1)^n \le 1\) for all \(n\). It is not a convergent sequence itself. However, it has subsequences:

• The subsequence of even terms: \(a_{2k} = (-1)^{2k} = 1\) for \(k \ge 1\). This subsequence is \(1, 1, 1, ...\), which converges to 1.
• The subsequence of odd terms: \(a_{2k+1} = (-1)^{2k+1} = -1\) for \(k \ge 0\). This subsequence is \(-1, -1, -1, ...\), which converges to -1.

This example illustrates how a bounded sequence that does not converge itself still contains convergent subsequences, as guaranteed by the Bolzano-Weierstrass theorem. This theorem is a critical result when studying the properties of sequences in real analysis.

Therefore, based on the Bolzano-Weierstrass theorem, every bounded sequence has a convergent subsequence.