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.