Rational Numbers form Subset of Real Numbers

Theorem
The set $\Q$ of rational numbers forms a subset of the real numbers $\R$.

Proof
Let $x \in \Q$, where $\Q$ denotes the set of rational numbers.

Consider the rational sequence:
 * $x, x, x, \ldots$

This sequence is trivially Cauchy.

Thus there exists a Cauchy sequence $\eqclass {\sequence {x_n} } {}$ which is identified with a rational number $x \in \Q$ such that:

So by the definition of a real number:
 * $x \in \R$

where $\R$ denotes the set of real numbers.

Thus, by definition of subset:
 * $\Q \subseteq \R$