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The problem includes three sequences:

- \( \{a_n\} = \left\{ \frac{1}{\ln(n+1)} \right\} \),
- \( \{b_n\} = \{\sin n\} \),
- \( \{c_n\} = \{\ln n\} \).

The task is to select all correct statements about these sequences.

Let's analyze the given sequences:

1. **Sequence \( b_n \cdot c_n \)**:  
   - \( b_n = \sin n \) is bounded because the sine function oscillates between \(-1\) and \(1\).  
   - \( c_n = \ln n \) grows without bound as \( n \to \infty \).  
   - Therefore, \( b_n \cdot c_n \) is unbounded as \( \ln n \) increases, even though \( \sin n \) is bounded.

2. **Sequence \( \frac{1}{a_n \cdot c_n} \)**:
   - \( a_n = \frac{1}{\ln(n+1)} \) tends to zero as \( n \to \infty \), meaning it becomes very small.
   - \( c_n = \ln n \) grows without bound.
   - Therefore, \( a_n \cdot c_n \sim \frac{\ln n}{\ln(n+1)} \approx 1 \) for large \(n\), so this product is bounded, making \( \frac{1}{a_n \cdot c_n} \) bounded as well.

3. **Sequence \( a_n \cdot b_n \)**:
   - \( a_n \) tends to zero as \( n \to \infty \), and \( b_n = \sin n \) is bounded.
   - Hence, \( a_n \cdot b_n \) tends to zero, making it infinitesimal (infinitely small) as \( n \to \infty \).

4. **Sequence \( a_n \cdot b_n \)** (again):
   - From the previous analysis, this sequence is infinitesimally small, so it's not bounded but tends to zero.

5. **Sequence \( \frac{1}{a_n} \)**:
   - Since \( a_n \) tends to zero as \( n \to \infty \), \( \frac{1}{a_n} \) grows without bound, meaning this sequence is unbounded.

In summary, the correct statements are:

- \( \{ \frac{1}{a_n \cdot c_n} \} \) is bounded.
- \( \{ a_n \cdot b_n \} \) is infinitesimally small.
- \( \{ \frac{1}{a_n} \} \) is unbounded.

Would you like more details on any of these points?

Here are 5 questions you might explore further:
1. How does the behavior of logarithmic sequences like \( \ln n \) compare to linear or polynomial growth?
2. Why does \( \sin n \) remain bounded while \( \ln n \) grows indefinitely?
3. What conditions make a sequence infinitesimally small?
4. Can a sequence that grows slowly still be considered unbounded?
5. How does the combination of a bounded and unbounded sequence affect the product?

**Tip**: When dealing with sequences involving logarithms, always check their behavior as \( n \to \infty \), as they tend to grow slower than polynomial or exponential terms.

The problem includes three sequences: an = {1 / ln(n+1)}, bn = {sin n}, cn = {ln n}. Choose all correct statements.

Explore a detailed analysis of three sequences: 1/ln(n+1), sin(n), and ln(n). This problem covers key concepts in sequence behavior, including boundedness and asymptotic growth. Ideal for undergraduate students studying calculus and advanced mathematics.

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