Real Multiplication is Closed

Theorem

The operation of multiplication on the set of real numbers $\R$ is closed:

$\forall x, y \in \R: x \times y \in \R$


Proof

From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.


Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes.

From the definition of real multiplication, $x \times y$ is defined as:

$\eqclass {\sequence {x_n} } {} \times \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n \times y_n} } {}$


We have that:

$\forall i \in \N: x_i \in \Q, y_i \in \Q$

therefore $x_i \times y_i \in \Q$.

So it follows that:

$\eqclass {\sequence {x_n \times y_n} } {} \in \R$

$\blacksquare$


Sources

  • 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
  • 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$
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