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You have uploaded an image from a textbook containing problems related to sequences and their convergence. The problems presented are:

1. **Problem 2**: Let \( x_n \to x \). Assume that \( x_n \geq 0 \) for all \( n \). Then show that \( x \geq 0 \).

2. **Problem 3**: Let \( a_n \leq b_n \) for \( n \in \mathbb{N} \). Assume that \( a_n \to a \) and \( b_n \to b \). Show that \( a \leq b \).

3. **Problem 4**: Let \( a \leq x_n \leq b \) for \( n \in \mathbb{N} \). If \( x_n \to x \), show that \( a \leq x \leq b \).

4. **Problem 5**: Let \( (x_n) \) and \( (y_n) \) be convergent. Let \( s_n := \min \{ x_n, y_n \} \) and \( t_n := \max \{ x_n, y_n \} \). Are the sequences \( (s_n) \) and \( (t_n) \) convergent?

5. **Problem 6**: Show that the set of bounded (real) sequences forms a real vector space.

6. **Problem 7**: True or False: If \( (x_n) \) and \( (x_n y_n) \) are bounded, then \( (y_n) \) is bounded.

Would you like a solution for any of these problems? I can guide you through the reasoning or provide a full solution!

Here are some follow-up questions you might want to explore:

1. How can we prove that the limit of a non-negative sequence remains non-negative?
2. What theorem or property supports the idea that limits preserve inequalities?
3. How does the boundedness of sequences affect their convergence behavior?
4. Why is it important to prove that a set of sequences forms a vector space?
5. How do operations like min and max affect the convergence of sequences?

**Tip**: When working with sequences, remember that the limit of a sequence inherits key properties of the terms (e.g., bounds, sign). This can be useful in many proofs!

The user submitted an image containing math problems related to sequences and their convergence.

Explore problems on sequences and their convergence, including proofs for non-negative limits, inequalities, and bounded sequences. Suitable for undergraduate-level students studying real analysis.

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