# Rational Number is Real Number

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## 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 $\eqclass {\sequence {x_n} } {}$ in $\Q$ such that:

- $x = \eqclass {\sequence {x_n} } {}$

So by the definition of a real number:

- $x \in \R$

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

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory