Let's analyze and solve both of the problems from the image:

Problem 4:

Prove that: $[0, 1] = \bigcap_{n=1}^{\infty} \left( -\frac{1}{n}, 1 + \frac{1}{n} \right)$

Solution:

• First, observe that the intersection is over open intervals $\left( -\frac{1}{n}, 1 + \frac{1}{n} \right)$, which are defined for each $n \in \mathbb{N}$. The intervals expand as $n$ increases.
• For $n=1$, the interval is $\left(-1, 2\right)$, which includes all numbers between $-1$ and $2$.
• For $n=2$, the interval is $\left(-\frac{1}{2}, \frac{3}{2}\right)$, which is smaller than the previous interval.
• As $n$ increases, the intervals shrink and approach $[0, 1]$. More formally, the endpoints of the intervals, $-\frac{1}{n}$ and $1 + \frac{1}{n}$, approach 0 and 1 as $n \to \infty$.

Thus, the intersection of all these intervals for $n = 1$ to infinity will be the closed interval $[0, 1]$. Therefore, we have: $[0, 1] = \bigcap_{n=1}^{\infty} \left( -\frac{1}{n}, 1 + \frac{1}{n} \right)$

Problem 5:

Prove that if $A$ is a family of sets and $B \in A$, then $B \subseteq \bigcup_{A \in A} A$.

Solution:

• Let $A$ be a family of sets, and $B \in A$.
• The union $\bigcup_{A \in A} A$ is the set of all elements that belong to at least one of the sets in $A$. Since $B \in A$, all elements of $B$ are also elements of the union.
• Formally, for every $x \in B$, since $B \in A$, we have $x \in B \subseteq \bigcup_{A \in A} A$.

Therefore, $B \subseteq \bigcup_{A \in A} A$.

Do you want further details on these proofs or have any questions? Here are five related questions to deepen understanding:

1. What is the significance of using open intervals in Problem 4?
2. Can you generalize Problem 4 for any interval $[a, b]$?
3. How would the proof in Problem 5 change if $B \notin A$?
4. Can you give an example of a family of sets to illustrate Problem 5?
5. What are some practical applications of intersections and unions of sets in real life?

Tip: In proofs involving intervals, always check how the boundaries behave as $n$ changes to ensure the correct interval limits.