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"Given any positive number $y$, no matter how large, and any positive number $x$, no matter how small, one can add $x$ to itself sufficiently many times so that the result exceeds $y$ (i.e., $nx>y$ for some $n \in \mathbb{N}$)."

There exists $x,y \in \mathbb{R}$ such that $x \neq 0$ and $\frac{x}{y}$ is a real number. By the Archimedean property (that the natural numbers have no bound), we can choose an $n \in \mathbb{N}$ such that $n > \frac{y}{x}$. Then by multiplying both sides by $x$, we have that $nx > y$.

Is the above statement enough reasoning that it is true?

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  • $\begingroup$ The first $\frac{x}{y}$ should be $\frac{y}{x}$, but yeah, other than that, it looks fine. I actually learned the Archimedean Property as "If $x\in\mathbb{R}^+$ and $y\in\mathbb{R}$, then there is some positive integer $n$ such that $nx>y$." So the first paragraph looks exactly like the Archimedean Property to me. $\endgroup$ – Sujaan Kunalan Sep 13 '14 at 23:56
  • $\begingroup$ It's pretty much supposed to be the Archimedean Property. We're just supposed to be using the Archimedean Property to prove a couple of axioms. $\endgroup$ – knerd Sep 14 '14 at 0:01

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