The question in the image asks to prove the following set equality:

[ \bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right] = (0, 1] ]

Proof Outline:

We need to show two inclusions:

1. Inclusion $\subseteq$: First, we want to prove that: [ \bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right] \subseteq (0, 1] ]

   • Take any $x \in \bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right]$.
   • By the definition of the union, there exists some $n \in \mathbb{N}$ such that $x \in \left[\frac{1}{n}, 1\right]$. This means that $\frac{1}{n} \leq x \leq 1$.
   • Since $\frac{1}{n} > 0$ for all $n \in \mathbb{N}$, we have $x > 0$ and $x \leq 1$.
   • Therefore, $x \in (0, 1]$.

   Thus, $\bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right] \subseteq (0, 1]$.

2. Inclusion $\supseteq$: Next, we show that: $(0, 1] \subseteq \bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right]$

   • Take any $x \in (0, 1]$. We need to show that $x \in \left[\frac{1}{n}, 1\right]$ for some $n \in \mathbb{N}$.
   • Since $x > 0$, we can find an $n \in \mathbb{N}$ such that $\frac{1}{n} < x$. This is possible because $\frac{1}{n} \to 0$ as $n \to \infty$.
   • Therefore, $x \in \left[\frac{1}{n}, 1\right]$ for sufficiently large $n$.
   • This shows that any $x \in (0, 1]$ is also in $\bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right]$.

Thus, we have shown both inclusions: [ \bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right] = (0, 1] ]

Do you want further details or have any questions?

Related Questions:

1. What is the difference between a closed interval $[a, b]$ and an open interval $(a, b)$?
2. Can you explain how the union of sets works in more detail?
3. How can we rigorously define the natural numbers, $\mathbb{N}$, used in this proof?
4. What would change if the upper bound were open, i.e., $\bigcup_{n \in \mathbb{N}} \left[\frac{1}{n}, 1\right)$?
5. How does the concept of limits play a role in this proof?

Tip:

Understanding the behavior of intervals and how unions work in set theory is essential for more advanced topics in real analysis, particularly when dealing with open and closed sets.