1. **Problem Statement:** We analyze the sets $A_1, A_2, A_3, A_4, A_5$ defined as: - $A_1 = \left\{ \frac{3n+1}{4n-1} \mid n \in \mathbb{N}^* \right\}$ - $A_2 = \left\{ \frac{1}{n} - \frac{1}{n^2} \mid n \in \mathbb{N}^* \right\}$ - $A_3 = \left\{ \frac{1}{x} \mid x \in [1,2] \right\}$ - $A_4 = \left\{ x^2 + 1 \mid x \in [0,2] \right\}$ - $A_5 = \left\{ (-1)^n + \frac{1}{n} \mid n \in \mathbb{N}^* \right\}$ For each set, we will: 1) Determine if it is bounded above, below, or both. 2) Find supremum and infimum. 3) Determine existence of maximum and minimum values. --- 2. **Set $A_1$ Analysis:** - Expression: $a_n = \frac{3n+1}{4n-1}$ - As $n \to \infty$, $a_n \to \frac{3}{4} = 0.75$ (limit of ratio of leading coefficients). - Compute first terms: - $n=1: \frac{3(1)+1}{4(1)-1} = \frac{4}{3} \approx 1.333$ - $n=2: \frac{7}{7} = 1$ - $n=3: \frac{10}{11} \approx 0.909$ - Sequence is decreasing and bounded below by $0.75$. **Boundedness:** Bounded above by $\frac{4}{3}$ (first term), bounded below by $0.75$. **Supremum:** $\sup A_1 = \frac{4}{3}$ (attained at $n=1$). **Infimum:** $\inf A_1 = \frac{3}{4}$ (limit, not attained). **Max:** Exists and equals $\frac{4}{3}$. **Min:** Does not exist (no term equals $0.75$). --- 3. **Set $A_2$ Analysis:** - Expression: $a_n = \frac{1}{n} - \frac{1}{n^2} = \frac{n-1}{n^2}$ - As $n \to \infty$, $a_n \to 0$. - Compute first terms: - $n=1: 1 - 1 = 0$ - $n=2: \frac{1}{2} - \frac{1}{4} = \frac{1}{4} = 0.25$ - $n=3: \frac{1}{3} - \frac{1}{9} = \frac{2}{9} \approx 0.222$ - Sequence increases from 0 to a maximum at $n=2$ then decreases. **Boundedness:** Bounded above by $0.25$, bounded below by $0$. **Supremum:** $\sup A_2 = 0.25$ (attained at $n=2$). **Infimum:** $\inf A_2 = 0$ (attained at $n=1$). **Max:** Exists and equals $0.25$. **Min:** Exists and equals $0$. --- 4. **Set $A_3$ Analysis:** - Expression: $a_x = \frac{1}{x}$ for $x \in [1,2]$ - $a_x$ is continuous and decreasing on $[1,2]$. **Boundedness:** Bounded above by $1$ (at $x=1$), bounded below by $\frac{1}{2} = 0.5$ (at $x=2$). **Supremum:** $\sup A_3 = 1$ (attained at $x=1$). **Infimum:** $\inf A_3 = 0.5$ (attained at $x=2$). **Max:** Exists and equals $1$. **Min:** Exists and equals $0.5$. --- 5. **Set $A_4$ Analysis:** - Expression: $a_x = x^2 + 1$ for $x \in [0,2]$ - $a_x$ is continuous and increasing on $[0,2]$. **Boundedness:** Bounded above by $2^2 + 1 = 5$, bounded below by $1$ (at $x=0$). **Supremum:** $\sup A_4 = 5$ (attained at $x=2$). **Infimum:** $\inf A_4 = 1$ (attained at $x=0$). **Max:** Exists and equals $5$. **Min:** Exists and equals $1$. --- 6. **Set $A_5$ Analysis:** - Expression: $a_n = (-1)^n + \frac{1}{n}$ - For even $n$: $a_n = 1 + \frac{1}{n} \to 1$ from above. - For odd $n$: $a_n = -1 + \frac{1}{n} \to -1$ from above. **Boundedness:** Bounded above by $2$ (at $n=2$), bounded below by $-1$ (limit). **Supremum:** $\sup A_5 = 2$ (attained at $n=2$). **Infimum:** $\inf A_5 = -1$ (limit, not attained). **Max:** Exists and equals $2$. **Min:** Does not exist. --- **Summary Table:** | Set | Bounded Above | Bounded Below | Supremum | Infimum | Max Exists | Min Exists | |------|---------------|---------------|----------|---------|------------|------------| | $A_1$ | Yes | Yes | $\frac{4}{3}$ | $\frac{3}{4}$ | Yes | No | | $A_2$ | Yes | Yes | $0.25$ | $0$ | Yes | Yes | | $A_3$ | Yes | Yes | $1$ | $0.5$ | Yes | Yes | | $A_4$ | Yes | Yes | $5$ | $1$ | Yes | Yes | | $A_5$ | Yes | Yes | $2$ | $-1$ | Yes | No |