The problem includes several mathematical exercises, mainly concerning inequalities, sequences, convergence, and limits. I will analyze and solve the questions step by step.

1) Inequality:

Prove that $(\forall n \in \mathbb{N}, n \geq 1): (1 + n^r) \geq 1 + nr$.

This is a basic inequality likely derived from the binomial theorem or convexity arguments. Here's a quick breakdown:

• For $r \geq 1$, use induction or differentiation to demonstrate this inequality.
  Would you like a formal proof of this?

2) Sequences Definitions:

We are given two sequences:

• $u_n = \left(1 + \frac{1}{n}\right)^n$,
• $v_n = \left(1 - \frac{1}{n}\right)^{-n}$.

a) Show that $\ln(u_n) = \int_0^1 \frac{1}{1 - t/n} dt$:
This involves expanding $u_n$ in logarithmic form and then using integration substitution.

b) Show that $u_n$ is increasing:
Demonstrate $u_{n+1} \geq u_n$ by showing the derivative of the sequence term is positive.

c) Show that $v_n$ is decreasing and convergent:
Prove $v_{n+1} \leq v_n$ and use properties of bounded sequences to argue convergence.

3) Sequence Comparison:

Prove that $(\forall n \in \mathbb{N}, n \geq 2): 0 \leq v_n - u_n \leq \frac{v_n}{n}$.
This requires bounding the difference between $v_n$ and $u_n$ using properties from part 1 and sequence convergence. The key is careful estimation.

4) Limit of $u_n$ and $v_n$:

Define $L$ as the common limit of $(u_n)$ and $(v_n)$.

a) Show $L \geq \frac{9}{4}$:
Use inequalities, the limit property, and bounds derived earlier to demonstrate this result.

b) Determine the value of $L$:
It is likely $L = e$ (Euler's number), as $u_n$ and $v_n$ converge to $e$ based on their definitions.

Would you like me to start solving each part in order or focus on a specific section? Let me know!

Follow-up questions to deepen understanding:

1. Why does $(1 + \frac{1}{n})^n$ converge to $e$?
2. What are the key properties of increasing or decreasing sequences?
3. How do integrals help us prove limits and bounds in sequences?
4. How does inequality $0 \leq v_n - u_n \leq \frac{v_n}{n}$ contribute to proving convergence?
5. How can we calculate exact sequence limits using tools like Taylor expansions?

Tip: Always confirm monotonicity (increasing or decreasing behavior) when working with sequence convergence.