In the image, you are asked to determine the set $E$ and then to find the supremum (sup) and infimum (inf) of $E$. I'll guide you through the highlighted cases.

Highlighted Case 2:

$E = \left\{ \frac{(-1)^n + 1}{p} \mid (n,p) \in \mathbb{N}^* \times \mathbb{N}^* \right\}$

• Interpretation:
  • $n$ and $p$ are natural numbers, excluding zero (i.e., $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$).
  • $(-1)^n$ alternates between $1$ when $n$ is even and $-1$ when $n$ is odd.

Thus, the expression for elements of $E$ simplifies as follows:

• When $n$ is even ($(-1)^n = 1$), the element becomes: $\frac{1 + 1}{p} = \frac{2}{p}$
• When $n$ is odd ($(-1)^n = -1$), the element becomes: $\frac{-1 + 1}{p} = \frac{0}{p} = 0$

Thus, the set $E$ consists of $0$ and $\frac{2}{p}$ for every $p \in \mathbb{N}^*$. So, $E = \left\{ 0, \frac{2}{1}, \frac{2}{2}, \frac{2}{3}, \dots \right\} = \left\{ 0 \right\} \cup \left\{ \frac{2}{p} \mid p \in \mathbb{N}^* \right\}$

• Finding sup and inf:
  • The infimum (inf) is the smallest value in the set, which is clearly $0$.
  • The supremum (sup) is the largest value, which is the limit of $\frac{2}{p}$ as $p \to 1$. Therefore, $\sup E = 2$.

Highlighted Case 3:

$E = \left\{ \frac{x + 1}{x + 2} \mid x \in \mathbb{R} \text{ and } x \leq -3 \right\}$

• Interpretation:
  • The expression $\frac{x+1}{x+2}$ is defined for all real numbers $x \leq -3$.

Now, we analyze this function. Consider the behavior of $\frac{x+1}{x+2}$:

• As $x \to -\infty$, the fraction tends to $1$, but since $x \leq -3$, the values remain less than $1$.
• At $x = -3$: $\frac{-3 + 1}{-3 + 2} = \frac{-2}{-1} = 2$

Thus, the function is decreasing for $x \leq -3$, and its maximum occurs at $x = -3$ with the value $2$, and the minimum value approaches $1$ as $x \to -\infty$.

• Finding sup and inf:
  • The supremum is the largest value, which occurs at $x = -3$, so $\sup E = 2$.
  • The infimum is the smallest value, which is approached as $x \to -\infty$, so $\inf E = 1$.

Would you like to dive deeper into any of these calculations or need further clarification? Here are five related questions you could explore:

1. What is the role of $n$ and $p$ in the generalization of Case 2's set $E$?
2. Can you explain how the sequence $\frac{2}{p}$ behaves as $p$ increases?
3. How would the set $E$ change in Case 3 if $x$ was restricted to different intervals?
4. What is the significance of limits in determining the supremum and infimum?
5. How does the behavior of the function $\frac{x+1}{x+2}$ change for different values of $x$?

Tip: When finding the supremum and infimum, always consider the boundary behavior of the set or function, as well as how it behaves as it approaches those boundaries.