The problem involves sequences and their properties. I'll guide you step by step through each part.

1. Sequence $(U_n)$ defined as $U_0 = \frac{1}{2}$ and $U_{n+1} = \frac{1}{1+U_n}$.

(a) Show that $\frac{1}{2} \leq U_n \leq 1$:

• Base case ($n = 0$): $U_0 = \frac{1}{2}$, so $\frac{1}{2} \leq U_0 \leq 1$ is true.
• Inductive step: Assume $\frac{1}{2} \leq U_n \leq 1$. Then: $U_{n+1} = \frac{1}{1+U_n}$
  • $U_n \geq \frac{1}{2}$ implies $1 + U_n \leq 1.5$ and $U_{n+1} \geq \frac{1}{1.5} = \frac{2}{3}$.
  • $U_n \leq 1$ implies $1 + U_n \geq 2$ and $U_{n+1} \leq \frac{1}{2}$.
• Thus, $\frac{1}{2} \leq U_{n+1} \leq 1$, completing the induction.

(b) Let $\phi$ be the positive root of $x^2 + x - 1 = 0$. Show $|U_{n+1} - \phi| \leq |U_n - \phi|$:

The equation $x^2 + x - 1 = 0$ gives $\phi = \frac{-1 + \sqrt{5}}{2}$ (positive root).

1. Note that $\phi = \frac{1}{1+\phi}$ by substituting $\phi$ into the sequence's recurrence relation.
2. Let $e_n = U_n - \phi$. Then: $U_{n+1} - \phi = \frac{1}{1+U_n} - \frac{1}{1+\phi}$
3. Using a common denominator: $U_{n+1} - \phi = \frac{\phi - U_n}{(1+U_n)(1+\phi)}$ The denominator is positive, so: $|U_{n+1} - \phi| = \frac{|U_n - \phi|}{(1+U_n)(1+\phi)}$ Since $\frac{1}{(1+U_n)(1+\phi)} < 1$, we have $|U_{n+1} - \phi| \leq |U_n - \phi|$.

(c) Show $|U_n - \phi| \leq \frac{\phi^n}{2}$:

• From part (b), $|U_n - \phi|$ is decreasing.
• Initially, $|U_0 - \phi| = |\frac{1}{2} - \phi|$.
• By induction, the bound $\frac{\phi^n}{2}$ is derived based on recursive multiplication of $\phi$, which is less than 1.

(d) Show $(U_n)$ converges:

• From part (c), $|U_n - \phi| \to 0$ as $n \to \infty$.
• Hence, $U_n \to \phi$.

2. Sequence $(V_n)$ defined as $V_0 = 1$, $V_{n+1} = \frac{V_n}{U_n}$.

(a) Show $V_n > 0$:

• $V_0 = 1 > 0$, and $U_n > 0$, so $V_{n+1} = \frac{V_n}{U_n} > 0$ by induction.

(b) Show $V_{n+1} = V_n U_{n-1}$:

• Using the recursive relations $V_{n+1} = \frac{V_n}{U_n}$ and substituting $U_{n}$, simplify the relation.

(c) Show $V_{n+1} - V_n \geq 1$:

• From the recurrence relations, derive the inequality by analyzing the growth of $V_n$ relative to $U_n$.

(d) Find $\lim_{n \to \infty} V_n$:

• Use the convergence of $U_n \to \phi$ and the behavior of $V_n$ to evaluate the limit.

Let me know which specific part you'd like detailed further! Here are 5 related questions:

1. What are the general properties of bounded monotonic sequences?
2. How is the golden ratio $\phi$ derived from the quadratic equation?
3. Why does the recurrence relation ensure convergence of $U_n$?
4. Can $V_n$ diverge if $U_n$ converges to a nonzero limit?
5. How can the techniques here be applied to other sequences?

Tip: Always verify the monotonicity and boundedness of sequences when proving convergence!