Rational Numbers form Subset of Real Numbers

Theorem
Let $x$ be a rational number.

Then $x$ is also a real number.

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 $\left[\!\left[{\left \langle{x_n}\right \rangle}\right]\!\right]$ in $\Q$ such that:
 * $x = \left[\!\left[{\left \langle{x_n}\right \rangle}\right]\!\right]$

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

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