What is

$\bigcup_{n=3}^\infty \left[\frac{1}{n},1 - \frac{1}{n}\right]?$

Notation. The bracket notation represents a closed interval, so $[3, 5] = \{x | 3 \leq x \leq 5\}$ is the set of all real numbers greater than or equal to 3 and less than or equal to 5.

The $\bigcup$ symbol represents set union, so $\{1, 2, 3\}\bigcup \{2,3,4\} = \{1, 2, 3, 4\}.$ The indexing $\bigcup_{n=3}^\infty$ represents an union of infinitely many sets, i.e.

$\bigcup_{n=3}^\infty \left[\frac{1}{n},1 - \frac{1}{n}\right] = \left[\frac{1}{3},1-\frac{1}{3}\right] \bigcup \left[\frac{1}{4},1- \frac{1}{4}\right] \bigcup \left[\frac{1}{5},1- \frac{1}{5}\right] \bigcup \cdots$