| Definition 2.7.6 The Archimedean property states that whenever $x \in \mathbb{R}$, and $a>0$, there exists $n \in \mathbb{N}$ such that $n a>x$. This is not hard to believe. Just look at the number line. Imagine the intervals $[0, a),[a, 2 a),[2 a, 3 a), \cdots$. If $x<0$, you could consider $a$ and it would be larger than $x$. If $x \geq 0$, surely, it is reasonable to suppose that $x$ would be on one of these intervals, say $[p a,(p+1) a)$. This Archimedean property is quite important because it shows every fixed real number is smaller than some integer. It also can be used to verify a very important property of the rationsl numbers. Axiom 2.7.7 $\mathbb{R}$ has the Archimedean property. Theoren $2.7 .8$ Suppose $x1$. Then there exists an integer, $l \in$ $\mathcal{Z}$, such that $x<l<y .$ If $x$ is an integer, there is no integer $y$ satisfying $x<y<x+1 .$ Proof: Let $x$ be the smallest positive integer. Not surprisingly, $x=1$ but this can be proved. If $x<1$ then $x^{2}<x$ contradicting the assertion that $x$ is the smallest natural number. Therefore, 1 is the smsllest natural number. This shows there is no integer $y$, satisfying $x<y<x+1$ since otherwise, you could subtract $x$ and conclude $0<y-x<1$ for some integer $y-x$. |
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