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The text in the image consists of several mathematical statements in Russian, asking to "Choose the correct statements." These statements deal with limits, subsequences, and boundedness in the context of sequences. Below is the translated version:

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**Select the correct statements:**

1. Consider the set of partial limits of the sequence \(\{a_n\}\). Let the number \(M\) satisfy the conditions: \(M\) is a partial limit of the sequence \(\{a_n\}\). For any other convergent subsequence \(\{a_{n_k}\}\), if \(\lim a_{n_k} \geq M\), then the number \(M\) is called the lower limit of the sequence \(\{a_n\}\).

2. Let the sequences \(\{a_n\}\), \(\{\beta_n\}\), and \(\{\gamma_n\}\) be infinitesimal (i.e., tending to zero as \(n \to \infty\)) and bounded. Then the sequence \(\{b_n\}\), where \(b_n = a_n + \beta_n + \gamma_n\), is also infinitesimal.

3. Consider the sequences \(\{a_n\}\), \(\{b_n\}\), and \(\{c_n\}\). Let the sequence \(\{a_n\}\) converge, \(\lim a_n = a\), and \(\lim b_n = a\). If the terms of the sequences starting from some index satisfy the inequality \(a_n \leq c_n \leq b_n\), then the sequence \(\{c_n\}\) also converges to \(a\).

4. Let the sequences \(\{a_n\}\), \(\{\beta_n\}\), and \(\{\gamma_n\}\) be infinitesimal and bounded. Then the sequence \(\{b_n\}\), where \(b_n = a_n - a_n \cdot \frac{1}{\beta_n} + \gamma_n\), is also infinitesimal.

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To analyze these statements:

1. The first statement appears to describe the *lim inf* (lower limit) of a sequence. The definition seems correct but would need closer inspection to confirm the logical structure.
2. The second statement is valid because the sum of infinitesimal sequences is also infinitesimal.
3. The third statement is a variation of the Squeeze Theorem (also known as the Sandwich Theorem) and is correct.
4. The fourth statement involves a more complex expression, and we would need to verify if \(b_n = a_n - a_n \cdot \frac{1}{\beta_n} + \gamma_n\) remains infinitesimal under the given conditions.

Would you like more details on each statement, or should we focus on analyzing a specific one?

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Here are 5 related questions to deepen the understanding:

1. What are the key properties of partial limits for sequences?
2. How does the Squeeze Theorem guarantee convergence in sequences?
3. Why does the sum of infinitesimal sequences remain infinitesimal?
4. What is the significance of boundedness in sequences in relation to convergence?
5. How can we rigorously define and calculate the lower limit (\(\lim \inf\)) of a sequence?

**Tip:** When working with sequences and limits, always verify the behavior of subsequences to understand overall convergence.

Select the correct statements based on limits, subsequences, and bounded sequences, as provided in the image.

Analyze and select the correct statements involving limits, subsequences, infinitesimal sequences, and boundedness. This exercise explores key mathematical concepts such as partial limits, lower limits, and the Squeeze Theorem.

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