This appears to be an exam on Real Analysis with questions related to sequences, series, and uniform continuity. I'll help you with the problems one by one. Let me know which specific question you'd like to work on first or if you need solutions for all of them.

Would you like a detailed step-by-step solution for any specific problem?

Here are 5 related questions you might want to explore:

1. How do you prove that a sequence is Cauchy, and why is this definition important in analysis?
2. What are the common convergence tests for infinite series, and how do you decide which one to use?
3. What is the distinction between absolute and conditional convergence for a series?
4. Can you give an example of a uniformly continuous function that is bounded?
5. Why is uniform continuity different from pointwise continuity, and how does this relate to real-world applications?

Tip: Always clarify the assumptions of theorems you use in your proofs to ensure rigor and avoid gaps in reasoning.