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The image you uploaded asks to study the convergence of a sequence, providing several detailed steps. Here's a transcription and analysis of the mathematical content:

### Problem Statement (Translated):
**Étudier la convergence d'une suite**  
Given a sequence defined by \( u_0 = 0 \) and for all \( n \geq 0 \), \( u_{n+1} = \sqrt{3u_n + 2} \).

1. **Prove that the sequence \( (u_n) \) is increasing**.
2. **Show that \( (u_n) \) is bounded**.
3. **Deduce the convergence of the sequence \( (u_n) \)**, and find its limit \( l \), as \( n \to \infty \).
4. **Solve the equation \( l = \sqrt{3l + 2} \) to determine the value of \( l \).**

### Let's break this down:

1. **Prove that the sequence \( (u_n) \) is increasing**:
   - To prove that \( (u_n) \) is increasing, you need to show that \( u_{n+1} \geq u_n \) for all \( n \geq 0 \). That is, prove that:
     \[
     \sqrt{3u_n + 2} \geq u_n.
     \]
   - Squaring both sides and simplifying should lead to an inequality that helps confirm this.

2. **Show that \( (u_n) \) is bounded**:
   - To prove that the sequence is bounded, you must find an upper bound \( M \) such that \( u_n \leq M \) for all \( n \geq 0 \).
   - Usually, you can do this by assuming that \( (u_n) \) converges, finding the potential limit \( l \), and checking that the sequence remains below this limit.

3. **Conclude the convergence of \( (u_n) \)**:
   - If a sequence is both increasing and bounded, by the Monotone Convergence Theorem, it must converge.
   
4. **Find the limit \( l \)**:
   - Solve the equation \( l = \sqrt{3l + 2} \) to determine the exact value of the limit.

Would you like a detailed solution for each step, or do you have any specific questions regarding the steps?

### 5 Relative Questions:
1. How do we check if squaring both sides in inequalities is valid?
2. What strategies can we use to find upper bounds of sequences like this?
3. Why does monotonicity help us conclude convergence?
4. How can we prove that the limit exists without solving the equation first?
5. Can the sequence \( (u_n) \) ever decrease for specific initial values?

### Tip:
When dealing with recursive sequences, always check for both monotonicity and boundedness. This combination usually guarantees convergence!

Étudier la convergence d'une suite donnée par la relation de récurrence u_0 = 0 et u_{n+1} = sqrt(3u_n + 2).

Explore the convergence of a recursive sequence defined by u_{n+1} = sqrt(3u_n + 2). Learn how to prove monotonicity, find boundedness, and solve for the limit using key mathematical concepts like inequalities and the Monotone Convergence Theorem.

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