# Real Multiplication is Closed

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## 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 $1$