$\mathbb{R}$ の部分集合族 $\{I_n\}_{n \in \mathbb{N}}$ を次の (1), (2) のように定める時
\begin{align}
\bigcup_{n = 1}^{\infty} I_n,\ \bigcap_{n = 1}^{\infty} I_n
\end{align}
を求めよ。
(1)
\begin{align}
I_n &= \left[- 2 + \frac{1}{2},2 – \frac{1}{n}\right]
\end{align}
(2)
\begin{align}
I_n = \left(a, b + \frac{1}{n}\right),\ (a, b \in \mathbb{R}, a < b)
\end{align}
(1)
\begin{align}
\bigcup_{n = 1}^{\infty} &= (-2, 2) \\
\bigcap_{n = 1}^{\infty} &= [-1, 1]
\end{align}
(2)
\begin{align}
\bigcup_{n = 1}^{\infty} &= (a, b + 1) \\
\bigcap_{n = 1}^{\infty} &= (a, b]
\end{align}
