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!