$\bigcup_{k=1}^{\infty} O_k\setminus\bigcup_{k=1}^\infty E_k \subset \bigcup_{k=1}^\infty[O_k\setminus E_k] $

Question

This is from Royden (chapter 2 Theorem 11). I don't understand one step in the proof. How do we know the following step holds:

$$\bigcup_{k=1}^{\infty} O_k\setminus\bigcup_{k=1}^\infty E_k \subset \bigcup_{k=1}^\infty[O_k\setminus E_k]. $$

The orginial image is here.

Answer 1

Take a point $x$ in the left side. Then $x$ is not in any of the sets $E_k$. Also, $x\in O_k$ for some $k$. For this $k$ we have $x \in O_k\setminus E_k$ so $x$ is in the right hand side.