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The image contains the statement and proof of Theorem 3.2.11 regarding sequences of positive real numbers and their limits. The theorem discusses the behavior of a sequence \((x_n)\) when the limit of the ratio \( \lim \frac{x_{n+1}}{x_n} = L \) exists, particularly when \( L < 1 \).

### Theorem 3.2.11:
Let \( (x_n) \) be a sequence of positive real numbers such that \( L := \lim \frac{x_{n+1}}{x_n} \) exists. If \( L < 1 \), then \( (x_n) \) converges, and \( \lim x_n = 0 \).

#### Proof Sketch:
- From Theorem 3.2.4, it follows that \( L \geq 0 \).
- Select \( r \) such that \( L < r < 1 \). Define \( \epsilon := r - L > 0 \).
- There exists a number \( K \in \mathbb{N} \) such that for all \( n \geq K \):
  \[
  \left| \frac{x_{n+1}}{x_n} - L \right| < \epsilon
  \]
  This implies:
  \[
  \frac{x_{n+1}}{x_n} < L + \epsilon = r
  \]
- Therefore, for all \( n \geq K \), we have:
  \[
  0 < x_{n+1} < x_n r < x_{n-1} r^2 < \cdots < x_K r^{n-K+1}
  \]
- Set \( C := x_K r^{K} \). Then, for \( n \geq K \), \( x_n < Cr^{n+1} \).
  Since \( 0 < r < 1 \), this implies \( \lim x_n = 0 \).

The proof concludes with the observation that the sequence tends to 0, using a comparison test and the fact that \( r^n \) tends to 0 as \( n \to \infty \).

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Would you like any details on specific parts of the proof or the theorem?

Here are 5 related questions to deepen your understanding:
1. What does Theorem 3.2.4 state, and how is it used in this proof?
2. Why is it necessary to choose \( r \) such that \( L < r < 1 \)?
3. How does the inequality \( \frac{x_{n+1}}{x_n} < r \) help in proving convergence of the sequence?
4. What role does the comparison sequence \( x_n < Cr^{n+1} \) play in concluding \( \lim x_n = 0 \)?
5. Can this theorem be extended to sequences with \( L = 1 \)? What happens in that case?

**Tip**: When dealing with sequences and their limits, try to find a comparison sequence that is easier to analyze, as it often simplifies the proof of convergence or divergence.

Theorem 3.2.11: Let (x_n) be a sequence of positive real numbers such that lim(x_(n+1)/x_n) = L exists. If L < 1, then (x_n) converges and lim(x_n) = 0.

Explore Theorem 3.2.11, which states that a sequence of positive real numbers converges if the limit of the ratio of consecutive terms is less than 1. The proof involves comparison tests and properties of limits, suitable for undergraduate-level real analysis courses.

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