Use the Bolzano-Weierstrass theorem to prove that every bounded sequence has a monotone
subsequence.
Hint: Consider a convergent subsequence {xnk }k∈N and let L := lim xnk . Distinguish two
cases: (1) xnk ≥ L for infinitely many k; (2) otherwise. Then use the definition of the limit
to find a monotone sub-subsequence.
Remark: In Section 22, Spivak first proves that every sequences has a monotone subse-
quence, and then obtains Bolzano-Weierstrass as a corollary. (Have a look!) Here you are
going in the other direction.

We are tasked with using the **Bolzano-Weierstrass theorem** to prove that every **bounded sequence** has a **monotone subsequence**. The Bolzano-Weierstrass theorem states that every bounded sequence has a **convergent subsequence**. We'll leverage this fact to find a monotone subsequence within a bounded sequence.

### Proof Outline:

1. **Given**: A bounded sequence \(\{x_n\}_{n \in \mathbb{N}}\), meaning there exists some \(M > 0\) such that \(|x_n| \leq M\) for all \(n \in \mathbb{N}\).
   
2. **By the Bolzano-Weierstrass theorem**: There exists a **convergent subsequence** \(\{x_{n_k}\}_{k \in \mathbb{N}}\), converging to some limit \(L \in \mathbb{R}\). Let:
   \[
   L = \lim_{k \to \infty} x_{n_k}
   \]
   
3. Now, distinguish two cases based on how the terms of \(\{x_{n_k}\}_{k \in \mathbb{N}}\) relate to the limit \(L\).

---

### Case 1: \(x_{n_k} \geq L\) for infinitely many \(k\)

In this case, there exists an infinite subsequence of \(\{x_{n_k}\}\) such that \(x_{n_k} \geq L\) for all sufficiently large \(k\). Let's define this subsequence as \(\{x_{m_k}\}_{k \in \mathbb{N}}\), which is such that:
\[
x_{m_k} \geq L \quad \text{for all } k \in \mathbb{N}.
\]
Since \(\{x_{m_k}\}\) converges to \(L\) (as it's a subsequence of \(\{x_{n_k}\}\)), and all its terms are greater than or equal to \(L\), the subsequence must be **monotonically decreasing** as \(k\) increases. Therefore, \(\{x_{m_k}\}\) is a monotone decreasing subsequence of the original sequence \(\{x_n\}\).

---

### Case 2: \(x_{n_k} < L\) for all but finitely many \(k\)

In this case, there exists an infinite subsequence of \(\{x_{n_k}\}\) such that \(x_{n_k} \leq L\) for all but finitely many \(k\). Let this infinite subsequence be \(\{x_{p_k}\}_{k \in \mathbb{N}}\), which is such that:
\[
x_{p_k} \leq L \quad \text{for all } k \in \mathbb{N}.
\]
Since \(\{x_{p_k}\}\) converges to \(L\) and all terms are less than or equal to \(L\), it follows that \(\{x_{p_k}\}\) is a **monotonically increasing** subsequence of the original sequence \(\{x_n\}\).

---

### Conclusion:

In both cases, we have constructed a monotone subsequence:
- In Case 1, a **monotonically decreasing subsequence** exists.
- In Case 2, a **monotonically increasing subsequence** exists.

Since every bounded sequence has a convergent subsequence (by Bolzano-Weierstrass), and every convergent subsequence can be analyzed as above, every bounded sequence must have a monotone subsequence.

This completes the proof.

---

Would you like more details about any of the steps? Here are 5 follow-up questions that can deepen your understanding:

1. Why is the Bolzano-Weierstrass theorem important for the existence of a convergent subsequence?
2. How does the boundedness of a sequence ensure that a convergent subsequence exists?
3. What happens if a sequence is not bounded? Can we still find a monotone subsequence?
4. How is the definition of a limit used to guarantee that the subsequences behave monotonically?
5. Can this result be extended to sequences in metric spaces?

**Tip**: When working with limits and subsequences, always think about the "tail behavior" of the sequence — what happens to terms as you move further along the sequence — this often guides you to construct useful subsequences.

Use the Bolzano-Weierstrass theorem to prove that every bounded sequence has a monotone subsequence.

Learn how to use the Bolzano-Weierstrass theorem to prove that every bounded sequence contains a monotone subsequence. This advanced real analysis problem covers important concepts like bounded sequences, convergent subsequences, and monotone behavior. Suitable for undergraduate students studying advanced calculus or real analysis.

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