Let us solve the exercise step-by-step based on the given problem.

Exercise 2 Explanation:

1. Definitions:

   • Two sequences $(u_n)$ and $(v_n)$ are called adjacent if:

     • One is increasing ($u_n$ is increasing),
     • The other is decreasing ($v_n$ is decreasing),
     • And $(u_n - v_n) \to 0$ as $n \to +\infty$.

   • If $(u_n)$ and $(v_n)$ are adjacent sequences such that $(u_n)$ is increasing and $(v_n)$ is decreasing, then for all $n \in \mathbb{N}$, $u_n \leq v_n$.

Question 1:

(a) Complete the sentence:

• "Toute suite croissante et majorée est convergente."

Explanation: This follows from the monotone convergence theorem, which states that a bounded, monotonic sequence is convergent.

(b) Complete the sentence:

• "Toute suite croissante et bornée est convergente."

Explanation: This is the same result, phrased differently. A bounded, increasing sequence will always converge.

Question 2:

Proposition:

"Deux suites adjacentes sont convergentes et elles ont la même limite."

Answer:

The statement is true. Here's why:

1. Since $(u_n)$ is increasing and bounded above (as $(v_n)$ is decreasing and acts as its upper bound), $(u_n)$ converges.
   Let $\ell = \lim_{n \to \infty} u_n$.

2. Similarly, since $(v_n)$ is decreasing and bounded below (as $(u_n)$ is increasing and acts as its lower bound), $(v_n)$ converges.
   Let $L = \lim_{n \to \infty} v_n$.

3. The fact that $(u_n - v_n) \to 0$ implies: $\ell - L = \lim_{n \to \infty} (u_n - v_n) = 0.$

4. Hence, $\ell = L$, so both sequences converge to the same limit.

Question 3:

Relate the curve $C$ of column $(I)$ to the curve $C'$ of column $(J)$, with justification.

Solution:

The relationship between the two curves can be justified as follows:

• Curve $C$ corresponds to the increasing sequence $(u_n)$, and curve $C'$ corresponds to the decreasing sequence $(v_n)$.
• The condition of adjacency ($u_n - v_n \to 0$) means the two curves "approach each other" as $n$ grows larger, ultimately converging to the same limit.

Thus, the curves represent the convergence of the two adjacent sequences to their common limit.

Final Answer Summary:

1. Question 1:
   (a) convergente.
   (b) bornée.

2. Question 2:
   The proposition is true, with a demonstration provided above.

3. Question 3:
   The curves $C$ (from $(I)$) and $C'$ (from $(J)$) represent $(u_n)$ and $(v_n)$, respectively, and they approach the same limit, as justified above.

Let me know if you would like detailed explanations or clarifications for any part!

Related Questions:

1. What is the monotone convergence theorem?
2. How can you prove that a sequence is bounded?
3. Why does the condition $(u_n - v_n) \to 0$ ensure convergence to the same limit?
4. What happens if $(u_n)$ and $(v_n)$ are not adjacent?
5. Can two sequences converge to different limits if $(u_n - v_n) \not\to 0$?

Tip: Always check if sequences are monotonic and bounded when proving convergence!