https://file.aiquickdraw.com/mathbot/file/1726349586691qag5mig3.jpg

The problem in the image defines a sequence \((u_n)\) such that:

- \(u_0 = 3\),
- \(u_{n+1} = 2 + \frac{1}{u_n}\) for all natural numbers \(n\).

The goal is to prove by induction that for all natural numbers \(n\), \(2 \leq u_n \leq 3\).

### Step 1: Base case (for \(n = 0\))
We are given that \(u_0 = 3\). Clearly, \(2 \leq u_0 \leq 3\), so the base case is verified.

### Step 2: Inductive Hypothesis
Assume that the statement is true for some \(n = k\). That is, assume that:
\[
2 \leq u_k \leq 3.
\]
We need to prove that the statement also holds for \(n = k+1\), i.e., \(2 \leq u_{k+1} \leq 3\).

### Step 3: Inductive Step (Prove for \(n = k+1\))
From the recurrence relation, we have:
\[
u_{k+1} = 2 + \frac{1}{u_k}.
\]
- Since \(u_k \geq 2\), we know that \(\frac{1}{u_k} \leq \frac{1}{2}\).
- Hence, \(u_{k+1} = 2 + \frac{1}{u_k} \leq 2 + \frac{1}{2} = 2.5\).
- Since \(u_k \leq 3\), we also have \(\frac{1}{u_k} \geq \frac{1}{3}\).
- Thus, \(u_{k+1} = 2 + \frac{1}{u_k} \geq 2 + \frac{1}{3} = 2.33\).

Therefore, we conclude that \(2 \leq u_{k+1} \leq 3\), which completes the inductive step.

### Conclusion
By mathematical induction, we have shown that for all natural numbers \(n\), \(2 \leq u_n \leq 3\).

Would you like further details on any part of this proof? Here are five related questions:

1. How does mathematical induction work?
2. Can you explain more about recurrence relations?
3. Why is the base case important in induction?
4. How do we establish bounds for a recursive sequence?
5. What other methods can be used to prove inequalities in sequences?

**Tip:** In induction, always ensure that both the base case and the inductive step are properly proven to fully establish the result.

The sequence (u_n) is defined by u_0 = 3 and, for all natural numbers n, u_{n+1} = 2 + 1/u_n. Prove by induction that for all natural numbers n, 2 ≤ u_n ≤ 3.

Learn how to prove the bounds of a recursive sequence using mathematical induction. The problem focuses on showing that 2 ≤ u_n ≤ 3 for a sequence defined by u_0 = 3 and u_{n+1} = 2 + 1/u_n. Suitable for advanced high school students.

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