I am having difficulty with a problem in one of my textbooks. It gives three definitions
$1$. A set is called a closed set if its complement is open.
$2$.The closure of a set E is the intersection of all Closed sets C of which E is a subset.
$3$.If is a sets closure is equal to $\mathbb R^n$, then it is dense.
The question asks to show that the set of rational numbers are a dense subset of the reals. This is a second year university textbook. Here is what I have so far: To show that the of set rational numbers is dense, it suffices to show that its closure is the set of Real Numbers. The closure of the set of rational numbers is defined to be the intersection of closed sets containing the rational numbers, and is thus the smallest closed set containing the rational numbers. A real number is defined to be the limit of a sequence of rational numbers, and therefore the smallest set that can contain all rational numbers (of which there exists a rational number between every two real numbers) as well as their boundary points (the irrational numbers) is the set of real numbers.