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.