Rational Numbers form Subset of Real Numbers
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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:
- $\Q \subseteq \R$
$\blacksquare$
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.31$. Definition
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory