# Proof

Note that since the rational Cauchy sequence of ones $$(1)_{n\in\mathbb N}$$ exists, it is also true that the real number $$1:=(1)_{n\in\mathbb N} + I$$ exists. It remains to be shown that the real number $$1$$ is neutral with respect to the multiplication of real numbers. Let $$x=(x_n)_{n\in\mathbb N}+I$$ be a real number. Because the rational Cauchy sequence $$(1)_{n\in\mathbb N}$$ is neutral with respect to the addition of rational Cauchy sequences, it follows that

$\begin{array}{ccll} x\cdot 1&=&((x_n)_{n\in\mathbb N}+I)\cdot ((1)_{n\in\mathbb N}+I)&\text{by definition of real numbers}\\ &=&((x_n)_{n\in\mathbb N}\cdot (1)_{n\in\mathbb N})+I&\text{by definition of multiplying real numbers}\\ &=&(x_n)_{n\in\mathbb N}+I&\text{because }(1)_{n\in\mathbb N}\text{ is neutral element of multiplication of rational Cauchy sequences}\\ &=&x&\text{by definition of real numbers}\\ \end{array}$

It remains to be shown that also the equation $$1\cdot x=x$$ holds for all $$x\in\mathbb R$$. It follows immediately from the commutativity of multiplying real numbers.

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013